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The Math Handbook
Everyday Math Made Simple
Richard Elwes
New York • London
© 2011 by Richard Elwes
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The language of mathematics
Primes, factors and multiples
Negative numbers and the number line
Arithmetic with fractions
The power of 10
Roots and logs
Percentages and proportions
Area and volume
Polygons and solids
Pythagoras’ theorem
Answers to quizzes
“I was never any good at mathematics.”
I must have heard this sentence from a thousand different people.
I cannot dispute that it may be true: people do have different strengths
and weaknesses, different interests and priorities, different opportunities
and obstacles. But, all the same, an understanding of mathematics is not
something anyone is born with, not even Pythagoras himself. Like all
other skills, from portraiture to computer programming, from knitting to
playing cricket, mathematics can only be developed through practice,
that is to say through actually doing it.
Nor, in this age, is mathematics something anyone can afford to ignore.
Few people stop to worry whether they are good at talking or good at
shopping. Abilities may indeed vary, but generally talking and shopping
are unavoidable parts of life. And so it is with mathematics. Rather than
trying to hide from it, how about meeting it head on and becoming good
at it?
Sounds intimidating? Don’t panic! The good news is that just a handful
of central ideas and techniques can carry you a very long way. So, I am
pleased to present this book: a no-nonsense guide to the essentials of the
subject, especially written for anyone who “was never any good at
mathematics.” Maybe not, but it’s not too late!
Before we get underway, here’s a final word on philosophy.
Mathematical education is split between two rival camps. Traditionalists
brandish rusty compasses and dusty books of log tables, while
modernists drop fashionable buzzwords like “chunking” and talk about
the “number line.” This book has no loyalty to either group. I have
simply taken the concepts I consider most important, and illustrated
them as clearly and straightforwardly as I can.
Many of the ideas are as ancient as the pyramids, though some have a
more recent heritage. Sometimes a modern presentation can bring a
fresh clarity to a tired subject; in other cases, the old tried and tested
methods are the best.
Richard Elwes
The language of mathematics
• Writing mathematics
• Understanding what the various mathematical symbols mean, and how to
use them
• Using BEDMAS to help with calculations
Let’s begin with one of the commonest questions in any mathematics
class: “Can’t I just use a calculator?” The answer is … of course you
can! This book is not selling a puritanical brand of mathematics, where
everything must be done laboriously by hand, and all help is turned
down. You are welcome to use a calculator for arithmetic, just as you
can use a word-processor for writing text. But handwriting is an
essential skill, even in today’s hi-tech world. You can use a dictionary
or a spell-checker too. All the same, isn’t it a good idea to have a
reasonable grasp of basic spelling?
There may be times when you don’t have a calculator or a computer to
hand. You don’t want to be completely lost without it! Nor do you want
to have to consult it every time a few numbers need to be added
together. After all, you don’t get out your dictionary every time you
want to write a simple phrase.
So, no, I don’t want you to throw away your calculator. But I would like
to change the way you think about it. See it as a labor saving device,
something to speed up calculations, a provider of handy shortcuts.
The way I don’t want you to see it is as a mysterious black box which
performs near-magical feats that you alone could never hope to do.
Some of the quizzes will show this icon
, which asks you to have a go
without a calculator. This is just for practice, rather than being a point of
Signs and symbols
Mathematics has its own physical toolbox, full of calculators, compasses
and protractors. We shall meet these in later chapters. Mathematics also
comes with an impressive lexicon of terms, from “radii” to “logarithms,”
which we shall also get to know and love in the pages ahead.
Perhaps the first barrier to mathematics, though, comes before these: it
is the library of signs and symbols that are used. Most obviously, there
are the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is interesting that once we
get to the number ten there is not a new symbol. Instead, the symbols
for 0 and 1 are recycled and combined to produce the name “10.”
Instead of having one symbol alone, we now have two symbols arranged
in two columns. Which column the symbol is in carries just as much
information as the symbol itself: the “1” in “13” does not only mean
“one,” it means “one ten.” This method of representing numbers in
columns is at the heart of the decimal system: the modern way of
representing numbers. It is so familiar that we might not realize what an
ingenious and efficient system it is. Any number whatsoever can be
written using only the ten symbols 0–9. It is easy to read too: you don’t
have to stop and wonder how much “41” is.
This way of writing numbers has major consequences for the things that
we do with them. The best methods for addition, subtraction,
multiplication and division are based around understanding how the
columns affect each other. We will explore these in depth in the coming
There are many other symbols in mathematics besides numbers
themselves. To start with, there are the four representing basic
arithmetical procedures: +, −, ×, ÷. In fact there are other symbols
which mean the same things. In many situations, scientists prefer a dot,
or even nothing at all, to indicate multiplication. So, in algebra, both ab
and a · b, mean the same as a × b, as we shall see later. Similarly,
division is just as commonly expressed by
as by a ÷ b.
This use of letters is perhaps the greatest barrier to mathematics. How
can you multiply and divide letters? (And why would you want to?)
These are fair questions, which we shall save until later.
Writing mathematics
Here is another common question:
“What is the point of writing out mathematics in a longwinded fashion?
Surely all that matters is the final answer?”
The answer is … no! Of course, the right answer is important. I might
even agree that it is usually the most important thing. But it is certainly
not the only important thing. Why not? Because you will have a much
better chance of reliably arriving at the right answer if you are in
command of the reasoning that leads you there. And the best way of
ensuring that is to write out the intermediate steps, as clearly and
accurately as possible.
Writing out mathematics has two purposes. Firstly it is to guide and
illuminate your own thought-processes. You can only write things out
clearly if you are thinking about them clearly, and it is this clarity of
thought that is the ultimate aim. The second purpose is the same as for
almost any other form of writing: it is a form of communication with
another human being. I suggest that you work under the assumption that
someone will be along shortly to read your mathematics (whether or not
this is actually true). Will they be able to tell what you are doing? Or is
it a jumble of symbols, comprehensible only to you?
Mathematics is an extension of the English language (or any other
language, but we’ll stick to English!), with some new symbols and
words. But all the usual laws of English remain valid. In particular, when
you write out mathematics, the aim should be prose that another person
can read and understand. So try not to end up with symbols scattered
randomly around the page. That’s fine for rough working, while you are
trying to figure out what it is you want to write down. But after you’ve
figured it out, try to write everything clearly, in a way that
communicates what you have understood to the reader, and helps them
understand it too.
The importance of equality
The most important symbol in mathematics is “=.” Why? Because the
number-one goal of mathematics is to discover the value of unknown
quantities, or to establish that two superficially different objects are
actually one and the same. So an equation is really a sentence, an
assertion. An example is “146 + 255 = 401,” which states that the
value on the left-hand side of the “=” sign is the same as the value on
the right.
It is amazing how often the “=” sign gets misused! If asked to calculate
13 + 12 + 8, many people will write “13 + 12 = 25 + 8 = 33.” This
may come from the use of calculators where the
button can be
interpreted to mean “work out the answer.” It may be clear what the line
of thought is, but taken at face value it is nonsense: 13 + 12 is not equal
to 25 + 8! A correct way to write this would be “13 + 12 + 8 = 25 +
8 = 33.” Now, every pair of quantities that are asserted to be equal
really are equal − a great improvement!
The “=” sign has some lesser-known cousins, which make less powerful
assertions: “<” and “>.” For example, the statement “A < B” says that
the quantity A is less than B. An example might be 3 + 9 < 13. Flipping
this around gives “B > A,” which says that B is greater than A, for
example, 13 > 3 + 9. The statements “A < B” and “B > A” look
different, but have exactly the same meanings (in the same way that “A
= B” and “B = A” mean essentially the same thing).
Other symbols in the same family are “≥” and “≤,” which stand for “is
greater than or equal to” and “is less than or equal to” (otherwise known
as “is at least” and “is at most”).
In coming chapters, we will look at techniques for addition, subtraction,
multiplication, division, and much else besides, which will allow us to
judge whether or not these types of assertion are true.
Now we will have a look at one of the hidden laws of mathematical
A profusion of parentheses
One thing you may see in this book, which you may not be used to, is
lots of brackets in among the numbers. Why is that? Rather than
answering that question directly, I’ll pose another. What is 3 × 2 + 1?
At first sight, this seems easy enough.
The trouble is that there are two ways to work it out:
a) 3 × 2 + 1 = 6 + 1 = 7
b) 3 × 2 + 1 = 3 × 3 = 9
Only one of these can be right, but which is it?
To avoid this sort of confusion, it is a good idea to use brackets to mark
out which calculations should be taken together. So the two above
would be written like this:
a) (3 × 2) + 1
b) 3 × (2 + 1)
Now both are unambiguous, and whichever one was intended can be
written without any danger of misunderstanding. In each case, the first
step is to work out the calculation inside the brackets.
The same thing applies with more advanced topics, such as negative
numbers and powers. In the coming chapters we shall see expressions
such as −42. But does this mean −(42), that is to say −16, or does it
mean (−4)2, which as we shall see in the theory of negative numbers, is
actually + 16?
You might protest that I haven’t answered the question at the start of the
last section. Without writing in any brackets, what is 3 × 2 + 1?
There is a convention which has been adopted to resolve ambiguous
situations like this. We can think of it as one of the grammatical laws of
mathematics. It is called BEDMAS (or sometimes BIDMAS or BODMAS).
It tells us the order in which the operations should be carried out:
Brackets Exponents Division Multiplication Addition Subtraction
If you prefer, “Exponents” can be replaced by “Indices,” giving BIDMAS
(or with “Orders,” giving BODMAS). All of these options are words for
powers, which we shall meet in a later chapter. (Unfortunately BPDMAS
isn’t quite as catchy.)
The point of this is that the order in which we calculate things follows
the letters in “BEDMAS.” In the case of 3 × 2 + 1, the two operations
are multiplication and addition. Since M comes before A in BEDMAS,
multiplication is done first, and we get 3 × 2 + 1 = 6 + 1 = 7 as the
correct answer.
When we come to −42, the two operations are subtraction (negativity,
to be pernickety) and exponentiation. Since E comes before S, the correct
interpretation is −(42) = −16.
Calculators use BEDMAS automatically: if you type in
you will get the answer 7 not 9.
Sum up The way we think about life comes across in the way We
talk and write about it. The same is true of mathematics. If you
want your thought-processes to be clear and accurate, then start by
focusing on the language you use!
1 Translate these sentences into mathematical symbols, and decide
whether the statement is true or false.
a When you add eleven to ten you get twenty-one.
b Multiplying two by itself gives the same as adding two to itself.
c When you subtract four from five you get the same as when you
divide two by itself.
d Five divided by two is at least three.
e Five multiplied by four is less than three multiplied by seven.
2 Put brackets in these expressions in two different Ways, and then,
work, out the two answers. (For example from 3 × 2 + 1, we get
(3 × 2) + 1 = 7 and
3 × (2 + 1) = 9.)
d 20 − 6 × 3
3 In each of the expressions in quiz 2, decide which is the correct
interpretation according to BEDMAS. (If it doesn’t matter, explain
4 As Well as BEDMAS, there is a convention that operations are
read from left to right. So 8 ÷ 4 ÷ 2 means (8 ÷ 4) ÷ 2 not 8 ÷
(4 ÷ 2). For which of addition, subtraction, multiplication, and
division is this rule necessary?
• Mastering simple sums
• Knowing how to “carry” and borrow
• Remembering shortcuts for mental arithmetic
Everyone knows what addition means: if you have 7 greyhounds and 5
chihuahuas, then your total number of dogs is 7 + 5. The difficulty is
not in the meaning of the procedure, but in calculating the answer. The
simplest method of all is to start at 7, and then add on 1 five times in
succession. This might be done by counting up from 7 out loud:
8,9,10,11,12, keeping track by counting up to 5 on your fingers.
But counting up is much too slow! When large numbers are involved,
such as 2789 + 1899, this technique would take several hours, and the
likelihood of slipping up somewhere is close to certain. So how can this
be speeded up? There are many different procedures which work well,
depending on the context, and the quantity and types of numbers we are
dealing with. We will have a look at several methods in this chapter.
The key thing is to be comfortable adding up the small numbers: those
between 1 and 9. Once you can do this without worrying about it, then
building up to larger and more complex sums becomes surprisingly easy.
The aim here is not just to arrive at the right answers, but to be able to
handle these types of calculation quickly and painlessly. If you feel you
could do with more practice, then set yourself five questions at a time
and work through them. Start as slowly as you like, and aim to build up
speed with practice.
When numbers grow up
It is no surprise that addition becomes trickier when it involves numbers
more than one digit long. So it is the length of the numbers that we have
to learn to manage next.
Suppose we are faced with the calculation 20 + 40. This seems easy. But
why? Because all that really needs to be done is to work out 2 + 4 and
then stick a zero on the end. In the same way, even three-, four-, or fivedigit numbers can be easy to handle: 3000 + 6000, for instance.
Things get slightly trickier when we have something like 200 + 900.
Here, although the question involves only three-digit numbers, the
answer steps up to four digits, just as 2 + 9 steps up from one digit to
Numbers with a lot of zeros are the first kind of longer numbers to get
used to.
Totaling columns
This chapter’s golden rule tells us how to tackle longer numbers: arrange
them in columns. The number “456,” for example, needs three columns.
It has a 6 in the units column, a 5 in the tens column, and a 4 in the
hundreds column:
Notice that we work along the columns from right to left, always
beginning with the units. (The reason for this backward approach will
become clear later on.) Now suppose we want to add 456 to another
number, say 123. The process is as follows. First write the two numbers
out in columns, with one under the other. Make sure that the units in the
top number are aligned with the units in the number below, and
similarly for the tens and hundreds columns.
With that done, all that remains is to add up the numbers in each
The art of carrying
Now we arrive at the moment where all the beautiful simplicity of the
previous examples turns into something a bit more complex. At this
stage the columns are no longer summed up individually, but start
affecting each other through a mystifying mechanism known as carrying.
I promise it isn’t as bad as it sounds!
Let’s start with an example: 44 + 28. What happens if we simply follow
the procedure described in the last section?
This is completely, 100%, correct! There is just one small worry: “sixtytwelve” is not the name of any number in English. (Saying it might
attract strange looks in the street.) So what is sixty-twelve in ordinary
language? A little reflection should convince you that the answer is
seventy-two. (In the French language, sixty-twelve, or soixante douze, is
in fact the name for seventy-two.) So, to complete the calculation, we
need to rewrite the answer in the ordinary way, as 72. What exactly is
going on in this final step? The answer is that the units column contains
12, which is too many. When we reduce 12 to 2, we are left with one
extra ten to manage. It is this 1 (ten) which is “carried” to the tens
Numbers are only ever carried leftwards: from the units column to the
tens, or from the tens to the hundreds. (This is the reason we always
work from right to left when adding numbers up.) Once we have grasped
this essential idea, we can speed up the process by doing all the carrying
as we go along.
So, let’s take another example: 37 + 68. Here we begin by adding up the
units column to get 15, which we can immediately write as 5 and carry
the leftover ten as an extra 1 to be included in the tens column. (We
write this as an extra 1 at the top of the column.) Then we add up the
tens column (including the carried 1) which produces 10. So we write
this as 0, and carry 1 to the hundreds column. Happily there is nothing
else in the hundreds column, so this is the end.
Lists of numbers
Whether it’s counting calories, or adding up shopping bills, addition is
probably our commonest use of numbers. But often the calculation needs
more than two numbers to be added together. The good news is that the
technique we learned in the last section transfers immediately to longer
lists of numbers. The rules are exactly the same as before: arrange the
numbers vertically, and then add each column in turn, starting on the
right, carrying when necessary. The only difference is that the number to
be carried this time might be larger than 1.
For example, to calculate 36 + 27 + 18 we set it up as:
This time the units column adds up to 21, so we write 1, and carry 2 to
the tens column. Then we add up the tens column as before, to get 8.
In your head: splitting numbers up
The addition techniques we have looked at so far work very well (after a
little practice). But they do have one downside: these are written
techniques. Often what we want is a way to calculate in our head,
without having to scuttle off to a quiet corner with a pen and paper.
Carrying can be tricky to manage in your head. Luckily there are other
ways to proceed.
If we want to add 24 to 51, one way to proceed is to split this up into
two simpler sums: first add on 20, and then add on another 4. Each of
these steps should be easy to do: 51 + 20 = 71 (because 5 + 2 = 7 in
the tens column). Then 71 + 4 = 75. The only challenge is to keep a
mental hold of the intermediate step (71 in this example).
Remember that you can choose which of the two numbers to split up. So
we could have done the previous example as 24 + 50 + 1. You might
find it better to split up the smaller of the two, but tastes vary.
Rounding up and cutting down
Imagine that a restaurant bill comes to £45 for food, with another £29
for drink. By now we have seen a few techniques we could use to tackle
the resulting sum: 45 + 29. But there is another possibility, which
begins by noticing that 29 is 1 less than 30. So, to make life easier, we
could round 29 up to 30. Then it is not hard to add 30 to 45 to get 75.
To complete the calculation, we just need to cut it back down by 1
again, to arrive at 74.
This trick of rounding up and cutting down will also work when adding,
say, 38 to 53. Instead of tackling the sum head-on, first round 38 up by
2, then add 40 to 53. To finish off, just cut that number back down by 2.
In some cases you might want to round up both numbers in the sum. For
example, 59 + 28 can be rounded up to 60 + 30, and then cut down by
a total of 3.
I think rounding up and cutting down is a good technique when the
units column contains a 7, 8 or 9 and splitting numbers up is better
when the units column contains a 1, 2 or 3. But it is up to you to decide
which approaches suit you best! So why not try both techniques?
Sum up Mathematics can teach us several techniques for addition
and subtraction. But all of them are based on familiarity with
the small numbers, 1 to 9.
After you have worked through these, come up with your own
examples if you want more practice. No calculators for this
1 In your head!
2 Numbers that grow longer a 30 + 40
b 5000 + 2000
c 800 + 300
d 7000 + 4000
e 30,000 + 90,000
3 Write in columns and add.
a 56 + 22
b 48 + 51
c 195 + 503
d 354 + 431
e 1742 + 8033
4 Mastering carrying a 14 + 27
b 36 + 38
c 76 + 85
d 127 + 344
e 245 + 156
5 Totaling longer lists a 14 + 22 + 23
b 27 + 44 + 16
c 26 + 47 + 28
d 19 + 28 + 17 + 29
e 57 + 66 + 38
6 Split these up, to Work, out in your head.
a 60 + 23
b 75 + 14
c 54 + 32
d 73 + 24
e 101 + 43
• Understanding how subtraction relates to addition
• Keeping a clear head when subtraction looks complicated
• Mastering quick methods to do in your head
As darkness is to light, and sour is to sweet, so subtraction is to
addition. As we shall see in this chapter, this relationship between
adding and subtracting is useful for understanding and calculating
subtraction-based problems. If you have 7 carrots, and you add 3, and
then you take away 3, you are left exactly where you started, with 7.
So subtraction and addition really do cancel each other out.
Getting started with subtraction
Subtraction is also known as taking away, for good reason. If you have 17
cats, of which 9 are Siamese, then the number of non-Siamese cats is
given by taking away the number of Siamese from the total number, that
is, by subtracting 9 from 17.
Now, there is one important theoretical way that subtraction differs from
addition: when we calculate 17 + 26, the answer is the same as for 26
+ 17. Swapping the order of the numbers does not make any difference
to the answer. But, with subtraction, this is no longer true: 26 − 17 is
not the same as 17 − 26. In a later chapter we will look at the concept
of negative numbers which give meaning to expressions such as 17 − 26.
In this chapter, we will stick to the more familiar terrain of taking
smaller numbers away from larger ones. (As it happens, extending these
ideas into the world of negative numbers is simple: while 26 − 17 is 9,
reversing the order gives 17 − 26, which comes out as −9. It is just a
matter of changing the sign of the answer. But we shall steer clear of this
for the rest of this chapter.)
The techniques for subtraction mirror the techniques for addition, with
just a little adjustment needed. And, as with addition, the first step is to
get comfortable subtracting small numbers in your head.
As ever, if you feel you could do with more practice, then set yourself
your own challenges in batches of five, starting as slowly as you like,
and aiming to build up speed and confidence gradually.
Longer subtraction
Now we move on to numbers which are more than just one digit long.
These larger calculations can be set up in a very similar way to addition
as this chapter’s golden rule tells us.
The first thing to do is to align the two columns one above the other,
making sure that units are aligned with units, tens with tens and so on.
Then the basic idea is just to subtract the lower number in each column
from the upper number. So to calculate 35 − 21 we would write this:
Taking larger from smaller: borrowing
What can go wrong with the procedure in the last section? Well, we
might face a situation like this:
The first step is to attack the units column. But this seems to require
taking 7 from 6, which cannot be done (at least not without venturing
into negative numbers, which we are avoiding in this chapter). So what
happens next? When we were adding, we had to carry digits between
columns. In subtraction, the opposite of carrying is borrowing. It works
like this: we may not be able to take 7 from 6, but we can certainly take
7 from 16. The way forward, therefore, is to rewrite the same problem
like this:
Notice that the new top row “forty-sixteen” is just a different way of
writing the old top row “fifty-six.” With this done, the old procedure of
working out each column individually, starting with the units, works
exactly as before.
What went on in that rewriting of the top row? We want to speed the
process up. Essentially, one ten was “borrowed” from the tens column
(reducing the 5 there to 4) and moved to the units column, to change the
6 there to 16. Usually, when writing out these sort of calculations, we
would not bother to write a little 1 changing the six to sixteen, since this
can be done in your head. But if it helps you to pencil in the extra 1,
then do it! It is usual, however, to change the 5 to 4 in the tens column.
To take another example, if we are faced with 94 − 36, the way to write
it out is like this:
Subtraction with splitting
This column-based method is very reliable and efficient. But, just as we
saw in the case of addition, it is not ideal when you want to calculate in
your head, instead of on paper. The first purely mental technique we
looked at for adding was splitting numbers up: to add 32 to 75, we split 32
up into 30 and 2, and then added these on separately, first 75 + 30 =
105, and then 105 + 2 = 107.
This approach works just as well with subtraction. (You might want to
remind yourself of how it worked for adding before continuing.)
In the context of subtraction, it is always the number being taken away
that gets split up. Suppose I know that there are 75 people in my office,
of whom 32 are men. I want to know how many women there are. The
calculation we need to work out is 75 − 32. The technique again
involves splitting the 32 up into 30 and 2. So first we take away 30 from
75, to get 45, and then subtract the final 2, to leave the final answer of
43 women. The aim is to complete the subtraction by splitting the
numbers up, without writing anything down. But, for practice, you
might want to write down the intermediate step, that is, 45 in the above
Rounding up and adding on
Another mental trick we learned for adding was rounding up and cutting
down. This works just as well for subtraction. The only thing to watch
out for is whether the numbers should be going up or down.
For example, to calculate 80 − 29, it might be convenient to round 29
up to 30. This gives us 50. It is in the final step that we need to take
care. Instead of cutting the answer down by 1 (as we did when adding),
this time we have subtracted 1 too many. So we have to add 1 back on,
to arrive at a final answer of 51.
Sum up Subtraction is the opposite of addition. Once you know
how to do one, it is just as easy to do the other!
1 Getting started a 9 − 6
c 12 − 5
d 17 − 9
e 16 − 8
2 Write in columns and subtract.
a 54 − 33
b 89 − 61
c 748 − 318
d 6,849 − 4,011
e 19,862 − 17,722
3 Get borrowing!
a 72 − 18
b 56 − 39
c 81 − 47
d 178 − 159
e 218 − 119
4 Split these up, to work out in your head.
a 60 − 23
b 75 − 14
c 54 − 32
d 73 − 24
e 101 − 43
5 Work out in your head, rounding up and adding on.
a 67 − 29
b 73 − 18
c 64 − 38
d 87 − 49
e 110 − 68
• Remembering your times tables
• Managing long multiplication
• Learning some tricks of the trade
What is multiplication? At the most basic level, it is nothing more than
repeated addition. If you have five plates, each holding four biscuits,
then the total number of biscuits is worked out by adding the numbers
on each plate. So 5 × 4 is shorthand for five 4s being added together: 4
+ 4 + 4 + 4 + 4.
This gives us our first way to calculate the answer: as long as we can add
4 to a number, we can work out 5 × 4 by repeatedly adding 4: 4, 8, 12,
16, 20. The fifth number (20) corresponds to the final plate added to the
biscuit collection, and so this is the answer.
We will see some slicker techniques shortly, but the perspective of
repeated addition is always worth holding in the back of your mind. It
also explains another word which is commonly used to describe
multiplication: “times.” The number 5 × 4 is the final result after 4 has
been added 5 times.
Multiplication is usually denoted by the times symbol, ×. If you are
working on a computer, though, often an asterisk * will play that role
(this was originally to prevent the times sign getting muddled up with
the letter X). When we get to more advanced algebra later, we will meet
other ways of writing multiplication, such as 4y or 4 · y.
As with addition (but not subtraction or division), the order of the
numbers does not matter. So 5 × 4 = 4 × 5, but the reason for this
may not be completely obvious. To see why this is true, we can arrange
the biscuits in a rectangular array as shown.
We can view this either as five columns, each containing four biscuits,
giving a total of 5 × 4, or alternatively as four rows, each containing
five biscuits, meaning that the total is 4 × 5. Of course this argument
extends to any two numbers, meaning that for any two numbers, call
them a and b, a × b = b × a.
Times tables
The trouble with the “repeated addition” approach is that it is not
practical for large numbers. To calculate 33 × 24 we would have to add
24s together 33 times. Most people have better ways of spending their
As with addition and subtraction, the key to more complex
multiplication is to get to grips with the smallest numbers: 1 to 9. What
this boils down to is times tables. For anyone hoping for an escape route,
I am sorry to say that there is none! But there are some ways by which
the pain can be reduced.
So here are some tips for mastering times tables:
• Firstly, remember the rule we saw above, that a × b = b × a.
Once you know 6 × 7 you also know 7 × 6!
• The two times table is just doubling, or adding the number to itself.
So 2 × 6 = 12 because 6 + 6 = 12.
• The four times table means doubling twice. So 4 × 6 = 24, because
6 + 6 = 12 and 12 + 12 = 24.
• The five times table has a simple rule: to multiply any number
(such as 7) by 5, first multiply it by 10 (to get 70) and then halve
the result (35).
• The nine times table also has a nice rule. Let’s look at it: 2 × 9 =
18, 3 × 9 = 27, 4 × 9 = 36, etc. There are two things to notice
here. Firstly, all the answers have the property that their two digits
add up to 9: 1 + 8 = 9, 2 + 7 = 9, and so on. What is more, the
first digit of the answer is always 1 less than the number being
multiplied by 9. So 2 × 9 = 18 begins with a 1, 3 × 9 = 27
begins with a 2, 4 × 9 = 36 begins with a 3, and so on. Putting
these together gives us our rule: To multiply a single-digit number
(such as 7) by 9, first reduce the number by 1 (to get 6). That is the
first digit of the answer. The second digit is the difference between
9 and the digit we have just worked out (in this case, 9 − 6 = 3).
Putting these together, the answer is 63.
The rules so far together cover a lot, but not everything. The first things
to be missed out are these four from the three times table: 3 × 3 = 9 3
× 6 = 18 3 × 7 = 21 3 × 8 = 24
It is also worth memorizing the square numbers separately, that is,
numbers multiplied by themselves (see Powers). Some of these are
covered by the rules so far. The remaining ones are: 6 × 6 = 36 7 × 7
= 49 8 × 8 = 64
Finally we get to the trickiest ones! These are the three multiplications
that people get wrong more than any others. It is definitely worth taking
some time to remember them: 6 × 7 = 42 6 × 8 = 48 7 × 8 = 56
Long multiplication
Even the most hard-working student can only learn times tables up to a
certain limit. These days, the maximum is usually ten, which seems a
sensible place to draw the line, and is the approach I’ve adopted here.
When I was at school, we learned them up to 12. The more ambitious
might want to push on, memorizing times tables up to 20.
Wherever you draw the line, to tackle multiplication beyond this
maximum, we need a new technique. It is time to put times tables to
Suppose we are asked to calculate 23 × 3. Unless we have learned our
three times table up to 23 (or our 23 times table up to 3), we need a new
approach. One option is to break multiplication down into repeated
addition: 23 + 23 + 23. But in the long run, a better method is to set
up the calculation in vertical columns:
To complete this, we multiply each digit of the upper number by 3, and
write it in the same column below the line. As long as we know our two
and three times tables, this is straightforward:
To calculate 41 × 4, we proceed exactly as before:
This time, the tens column produces a result of 16, and we have finished.
Just as for addition, the moment that multiplication seems to become
more complex is when the columns start interfering with each other, and
the dreaded “carrying” becomes involved again.
Well, as I hope became clear in the addition chapter, carrying is not as
confusing as you might think. In fact we have already seen some
carrying in this chapter. Above, when we calculate 41 × 4, the tens
column ended up with 16 in it. Of course this is too many, so it was
reduced to 6, and the 1 was carried to the hundreds column, though we
may not have noticed it happening.
To take another example, let us say we want to calculate 16 × 3. If we
just follow the rules above of multiplying each column separately, it
comes out as follows:
This leaves us with the correct answer, but expressed in an unusual way:
thirty-eighteen. So what is that? Thinking about it, the answer must be
What happens here is that the extra 1 ten from the units column gets
added to the 3 in the tens column.
As with addition, it is usual to write the carried digits at the top as we go
along. The crucial point to remember is: Carried digits get added (not
multiplied), to the next column, after that column’s multiplication has been
So, when written out, the above calculation would look like this:
The 4 comes from the fact that three times 1, plus the carried 1, is 4.
Here is another example:
We begin with the units column, where 6 × 7 = 42. So we write down
the 2 and carry the 4 to the next column:
Next, we tackle the tens column, where 7 × 2 = 14, and then we add
on the carried 4 to get 18:
(Technically, the final step involved writing down 8, and carrying 1 to
the hundreds column, where there is nothing else.)
Numbers march left
Which is the easiest times table? Apart from the completely trivial one
times table, the answer is the ten times table. Multiplying by 10 is
simple: you just have to copy the original number down, and then stick a
zero at the end. So 10 × 72 = 720.
To say the same thing in a different way: when writing the number in
columns of units, tens and hundreds, multiplying by 10 amounts to the
digits of the number each taking a step to the left. So the units move to
the tens column, the tens move to the hundreds column, and so on:
As always, any apparently “empty” columns actually have a 0 in them,
which is where the extra zero on the end comes from. This perspective,
of the digits stepping left when multiplied by 10, is the best one for
Another way to think of the same thing, is that in multiplying 72 by 10,
we begin at the units column, with 2 × 10, which would give 20, but
this means 0 in the units column, with 2 being carried to the tens
column. In the same way, the 7 is carried from the tens to the hundreds
column. This leftwards step, then, is nothing more than each digit being
carried, without change, straight to the next column to their left.
With this in mind, multiplying by 20 or 70 becomes as easy as
multiplying by 2 or 7. So 9 × 20 = 180, just because 9 × 2 = 18, and
then the digits take a step to the left.
This technique combines well with the previous section. When faced
with a calculation such as 53 × 30, we proceed exactly as for 53 × 3,
but placing a 0 in the units column, and shifting each subsequent digit
one column to the left:
Putting it all together
We nearly have the techniques in place to multiply any two numbers. All
that remains is to bring it all together. The critical insight at this stage is
this: multiplying some number, say 74, by 52 is the same as multiplying
it by 50, and separately multiplying it by 2, and then adding together the
two answers. Remember this chapter’s golden rule!
Why should this be? Suppose I am the door-keeper at a concert. The
entry charge is 52 pence. To make life easy, let’s suppose that everyone
pays with a 50p coin and two 1p coins. If 74 people come in, then how
much money have I received? The answer, of course, is the number of
customers times the price: 74 × 52 pence. But I decide to work it out
differently, and calculate the total I have received in 50p coins (74 ×
50), and then add that to the amount I have received in 1p coins (74 ×
2). Of course the answer should be the same, that is to say: 74 × 52 =
74 × 50 + 74 × 2.
The grid method
We can push this line of thought further. By exactly the same reasoning,
it is also true that 74 × 50 = 70 × 50 + 4 × 50 and similarly that 74
× 2 = 70 × 2 + 4 × 2. (Just alter the numbers in the concert
example!) This provides us with a way to calculate the answer to 74 ×
52, known as the grid method. We work inside a grid, with one of the two
numbers to be multiplied going along the top, and the other along the
left-hand side. Then each of the two is split up into their tens and units
Inside the grid, we then perform the resulting four multiplications:
The final stage is to add these four new numbers together, to arrive at
the final answer: 3500 + 200 + 140 + 8 = 3848.
The grid method easily extends to three-digit numbers. But it becomes
quite time-consuming, as we have to perform nine separate calculations.
For instance, to calculate 136 × 495 we split it up as follows:
All that remains is to fill in the gaps, and add them up.
The column method
I think the grid method for multiplication is an excellent way to get used
to multiplying larger numbers. So, if you are unsure of your foothold on
this sort of terrain, my suggestion is to persevere with the grid method
until you get comfortable with it.
Once you are used to the grid method, however, there is another step
you can take: the column method. This has the advantage of taking up
less space on the page, and less time, as it needs a much smaller number
of individual calculations.
Essentially the idea is to split up one of the two numbers into hundreds,
tens, and units, as occurs in the grid method, but not the other. This
amounts to calculating each row of the grid in one go. (With three-digit
numbers, this reduces the list of numbers to be added from nine to
three.) As its name suggests, we are back to working in columns instead
of grids. It works like this: to calculate 56 × 42 write the two numbers
in columns.
Next, ignore the “4,” and simply multiply 56 by 2, by the usual method
of “carrying”:
Then we swap: ignore the 2 in the 42 (and the new 112), and this time
multiply 56 by 40. Remember that this entails multiplying by 4, and
shifting the answer one step to the left:
The final stage is to add the two bottom lines together:
Sum up Build up multiplication step by step, starting with
repeated addition, until long multiplication is easy!
1 A times table test!
2 In columns
a 34 × 2
b 22 × 4
c 31 × 3
d 64 × 2
e 41 × 5
3 March to the left a 44 × 20
b 23 × 30
c 12 × 40
d 63 × 30
e 71 × 50
4 Multiplication in a grid a 34 × 21
b 45 × 34
c 62 × 45
d 71 × 123
e 254 × 216
5 Long multiplication in columns a 76 × 12
b 61 × 34
c 57 × 29
d 152 × 73
e 313 × 84
• Using times tables backward
• Remembering long division
• Understanding chunking
Just as subtraction is the opposite of addition, so division is the
opposite multiplication. More precisely, 24 ÷ 6 is the number of times
that 6 fits into 24. We could rephrase the question as “6 ×
= 24”;
by which number do we need to multiply 6 to get 24? What is this
useful for? Well, suppose I want to share a packet of 24 sweets among
6 salivating children. If each child is to get the same number of sweets
(seems a good idea—to avoid an almighty argument) then that number
must be 24 ÷ 6.
The usual symbol for division is “÷,” but computers often display it as
“.” Another way of representing division is as a fraction, so “24 ÷ 6,” “246”
all have exactly the same meaning.
Getting started with division
As usual, the starting point for division is to get used to working with
the small numbers, 1 to 9. In particular it is very useful to be able to
work backward from the times tables, and to be able to answer questions
like this: 6 ×
= 42. (This is the same as calculating 42 ÷ 6.)
When things don’t fit: remainders
When we are doing division with whole numbers, something rather
awkward can happen, something that we didn’t see with addition,
subtraction or multiplication. In the case of addition, for example, if you
start off with two whole numbers, then when you add them together,
you will produce another whole number. But with division, this can go
wrong. If we try to work out 7 ÷ 3, for example, we seem to get stuck. If
we know our three times table, then we know that 7 isn’t in it: the table
jumps from 2 × 3 = 6 to 3 × 3 = 9. So what can we do?
Let’s go back to the example of dividing up sweets between children.
Suppose we have 7 sweets to divide between 3 children. To avoid a
fight, we want each child to get the same number of sweets. How many
can they each have? With a little reflection, the answer is 2. That leaves
1 left over, which we can put back in the bag (or eat ourselves). We can
say that 7 divided by 3 is “2 with remainder 1.” We write that as: 7 ÷ 3
for short. Questions like this are a tougher test of your times tables! This
is how to tackle them.
• If we want to calculate 29 ÷ 6, the first thing to do is to go through
the six times table to find the last number in that list which is
smaller than (or equal to) 29. With a little reflection, we see that
number is 24.
• The next question is: 6 ×
= 24? The answer is 4. So 29 ÷ 6 is
equal to 4, with some remainder.
• The final step is to find out what that remainder is: it is the
difference between 29 and 24, which is 5. So the final answer is: 29
Sometimes it is best to leave the answer to a division question as a
remainder. But there are other options. To go back to the example
above, where 7 sweets were divided between 3 children, we had an
answer of 7 ÷ 3 = 2 r 1. One way to deal with the 1 remaining sweet is
to chop it into thirds, and give each child one third. In total then, each
child will have received
sweets, so
It is not hard to move between the language of remainders and fractions:
• Once we have arrived at 2 r 1, the main part of the answer (that’s
2) remains the same.
• Then the remainder (1) gets put on top of a fraction, with the
number we divided by (3) on the bottom, to give
So, to take another example, having worked out 29 ÷ 6 = 4 r 5, we can
express this as a fraction as
. It is dealing with remainders which gives
division its unique flavor.
The word “chunking” is a fairly new addition to the mathematical
lexicon, the sort of thing that might make traditional mathematics
teachers raise their eyebrows. All the same, many schools around the
world teach this method today. So what is chunking all about?
Actually, far from being something fancy and modern, chunking is an
ancient and very direct approach to division problems involving larger
numbers. It is just the word that is new!
Suppose we want to divide 253 by 11. The idea is to try to fit bunches of
11 inside 253, thereby breaking it up into manageable chunks. So the
smaller number (11) comes in bunches, and the larger number (253) gets
broken down into chunks. Got that?
Now, a bunch of ten 11s amounts to 110, and this certainly fits inside
253. In fact, it can fit inside twice, since twice 110 is 220 (but three
bunches comes to 330 which is too big).
So we have broken up 253 into two chunks of 110, which with have
been dealt with. The leftover is 253 − 220 which is 33. To continue, we
want to fit more 11s into this final chunk. Well, 11 can fit into 33 three
times. All in all then, we fitted 11 into 253 twenty times and then a
further three times. So 253 ÷ 11 = 23.
With chunking the key is to start by fitting in the largest bunch of 11s
(or whatever the smaller number is) that you can, whether that is
bunches of ten or a hundred. Doing this reduces the size of the leftover
chunk, making the remaining calculation easier.
You may find it helpful to make notes as you work, to keep track of the
chunks that have been dealt with, and the size of the leftover chunk.
Short division
What happens when the numbers involved are larger? Suppose we are
faced with a calculation like 693 ÷ 3. Chunking is one option, but when
the numbers are larger, it’s worth knowing a careful written method.
Division is set out in a different way from the column approach of
addition, subtraction and multiplication:
One reason for this change is that when doing addition and
multiplication we work from the right (from units to tens to hundreds).
In division, we work from the left, starting with the hundreds. The
reason for this swap will become apparent soon!
For now, the way to approach calculations such as the above is to start
with the hundreds column of the number inside the “box” (in this case,
693, known in the jargon as the “dividend”), and ask how many times 3
(the “divisor”) fits into it. That is to say, we begin by calculating 6 ÷ 3.
The answer of course is 2, so this is written above the 6, like this:
With this done, we move to the next step, which is to do the same thing
for the tens column, and then the units. After all this, the final answer
will be found written on the top of the “box”:
One thing to remember is that 0 divided by any other number is still 0.
So if we are working out 804 ÷ 4, when we reach the tens column, we
have to calculate 0 ÷ 4. This is 0. So working it through exactly as we
did above, we get:
Remainders go to work: carrying
As you might have feared, things do not always go quite as smoothly as
the last section suggests. What might go wrong?
Suppose a group of 5 friends group together to buy an old car for £350.
How much does each of them have to pay? The calculation we need to
do is 350 ÷ 5. We can set it up as before:
According to the previous section, the first step is to tackle the hundreds
column: 3 ÷ 5. But 5 doesn’t go into 3. The five times table begins: 0, 5,
10, 15, 20, … with 3 nowhere to be seen. So we’re stuck. What happens
The answer is we use our old friend “carrying,” albeit in a different guise
from before. Also, remember remainders: 5 fits into 3 zero times with
remainder 3. So we write a zero above the 3. But this leaves a leftover 3
in the hundreds column. This is carried to the tens column where it
becomes 30. Added to the 5 that is already there, we get 35 in the tens
column. That’s usually written like this:
Now we can carry on as before: since 35 ÷ 5 = 7:
So we arrive at an answer of 70.
What happened during this new step was that we essentially split up 350
in a new way. Instead of the traditional 3 hundreds, 5 tens and 0 units,
we rewrote it as 0 hundreds, 35 tens, and 0 units. With this done, the
calculation could proceed exactly as before.
Let’s take another example. Say 984 ÷ 4. As ever, the first thing to
tackle is the hundreds column, where we face 9 ÷ 4. This is slightly
different from the last example, where we had 3 ÷ 5. In that case, 5
could not fit into 3 at all; it was just too big. But this time 4 does fit into
9. The answer is 2, with a remainder of 1. This remainder gets carried to
the next column. The 2 is written above the 9. That takes us this far:
The next stage is to tackle the tens column, where we have 18 ÷ 4.
Once again, this doesn’t fit exactly, but gives an answer of 4 with
remainder 2. So the 4 gets written above the 8, and the remainder is
carried to the next column:
The final step is the units column, where we have 24 ÷ 4. That is 6. So
we have our final answer: 246.
Long division
There are few expressions in the English language that induce as much
horror as “long division.” In fact, it’s not so bad. Long division is
essentially the same thing as the short division we have just met. It’s just
a little bit longer.
The difference is that as the numbers involved become larger we may
have to carry more than one digit at a time to the next column. So
calculating the remainders becomes more cumbersome. Rather than
cluttering up the division, the remainders are written underneath
instead. So, if we wanted to calculate 846 ÷ 18, short division would
look like this:
while long division occupies a little more space:
What is the meaning of the column of numbers underneath?
Since 18 cannot divide the 8 in the hundreds column, we carry the 8 and
move on to the next column. The only difference is that we write the 84
underneath this time. Then 18 goes in to 84 four times, since 4 × 18 =
72, but 5 × 18 = 90 which is too big. So 4 is written on top, just as
before, and 72 is written below 84 and then subtracted from it to find
the remainder, 12.
If we were doing short division, 12 would be the number we carry to the
next column and stick in front of the next digit. But because we are
doing things underneath, we bring down the next digit from 846
(namely 6) and stick it on the end of the 12 to get 126. The last step is to
try to divide 126 by 18. A little chunking shows that 18 fits in exactly 7
times, so 7 is written on the top, to complete the calculation.
Dare you try long division? Don’t be put off by the numbers underneath:
if you’re not sure what you should be writing down there, try laying the
whole thing out as a short division, and doing any supplementary
calculations you need underneath. Remember: the working underneath
is intended to help you with the calculation, not to confuse you!
Sum up There are several methods for bringing division down to
earth. But even long division is manageable, once you have a
good grasp on remainders!
1 Times tables, backward!
= 12
= 30
= 27
= 64
= 63
2 Write out as remainders and as fractions.
a 11 ÷ 4
b 16 ÷ 6
c 24 ÷ 7
d 48 ÷ 5
e 59 ÷ 8
3 Chunking
a 96 ÷ 8
b 154 ÷ 7
c 279 ÷ 9
d 372 ÷ 6
e 8488 ÷ 8
4 Lay these out as short divisions.
a 864 ÷ 2
b 770 ÷ 7
c 903 ÷ 3
d 8482 ÷ 2
e 9036 ÷ 3
5 Short division
a 605 ÷ 5
b 426 ÷ 3
c 917 ÷ 7
d 852 ÷ 6
e 992 ÷ 8
6 Long division! Dare you. try it?
a 294 ÷ 14
b 270 ÷ 15
c 589 ÷ 19
d 1785 ÷ 17
e 1464 ÷ 24
Primes, factors and multiples
• Understanding prime numbers and why they are so important
• Being able to tell when one number is divisible by another
• Knowing how to break a number down into its basic components
Odd numbers, even numbers, prime numbers, composite numbers,
square numbers, … these are just a few of the different types of
numbers that mathematicians get incredibly excited about. What are all
these different sorts of number? Most of these terms refer to the
different ways that whole numbers are built out of others. This will
become clearer when we have met the most important numbers of all:
prime numbers.
Prime numbers
The definition of a prime number is simple: a prime number is a whole
number which cannot be divided by any other whole number (except 1
and itself). So, for example, 3 is prime because the only way to write 3
as two positive whole numbers multiplied together is as 3 × 1 (or 1 ×
3, which is essentially the same thing). On the other hand 4 is not prime
because 4 = 2 × 2.
A composite number essentially means a “non-prime” number, and 4 is
the first example. Similarly 5 is prime, but 6 is composite. (The numbers
0 and 1 are so special that they deserve categories of their own, and are
classed as neither prime nor composite.) The first 25 primes are: 2 3 5 7
11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
It was Euclid, in around 300 BC, who first proved that the list of primes
goes on forever. There is no largest prime number, and so people keep
finding bigger and bigger ones. It is a tough job though, as telling
whether a very large number is prime or composite is hard. The largest
prime known so far is 12,978,189 digits long!
The atoms of mathematics
Why do people get so excited about prime numbers? The reason they are
so important is that they are the fundamental blocks from which all
other numbers are built. Although 6 is not prime, it can be broken down
into primes as 3 × 2. Similarly 8 can be broken down as 2 × 2 × 2,
and 12 as 2 × 2 × 3. In this sense, prime numbers are like
mathematical atoms: everything else is built from them.
What is more, this chapter’s golden rule says a little bit more than this.
Not only can every number be broken down into primes, but there is
only one way to do it. So once we know that 1365 = 3 × 5 × 7 × 13,
for example, it follows that the only other ways to write 1365 as a
product of prime numbers are reorderings of this: 5 × 3 × 13 × 7, for
example. So we know automatically, without having to check, that 1365
≠ 5 × 5 × 5 × 11 (the symbol ≠ means “is not equal to”). This rule
goes by the grand title of The fundamental theorem of arithmetic.
Even and odd
Even numbers are those which appear in the two times table: 2, 4, 6, 8,
10, … Another way to say the same thing is that even numbers are those
which have 2 as a factor, meaning that 2 can divide into the number
exactly, without leaving a remainder. Yet another way to say the same
thing, is that the even numbers are the multiples of 2.
Odd numbers, of course, are the remaining numbers: the numbers which
do not have 2 as a factor.
Factor and multiple are opposite terms. To say that 15 is a multiple of 3 is
the same as saying that 3 is a factor of 15. Both statements mean that 3
can divide into 15 exactly, without leaving a remainder. In other words,
15 is in the three times table.
Divisibility tests
It is often useful to know whether or not a large number is a multiple of
a particular smaller number. For some small numbers this is so easy that
we can do it without thinking: • The multiples of 2 are exactly the even
numbers, meaning all the numbers that end in 2, 4, 6, 8 or 0.
• The multiples of 5 are the numbers that end in a 5 or a 0, such as
75 and 90.
• The multiples of 10 end in 0s, such as 80, 250, 16,700.
For other small numbers there are other tests, which are slightly subtler:
• You can tell whether or not a number is a multiple of 3 by adding up
its digits. If the total is a multiple of 3, then so was the original number.
So 117 is a multiple of 3, because 1 + 1 + 7 = 9, which is a multiple of
3. On the other hand 298 is not a multiple of 3, because 2 + 9 + 8 =
• A number is a multiple of 6 if it passes the tests for 2 and 3. So 528
is divisible by 6, since it is even, and 5 + 2 + 8 = 15, which is
divisible by 3. (Notice that the total of the digits does not have to
be divisible by 6.) • The test for divisibility by 9 is similar to the
test for 3: add up the digits, and if the result is a multiple of 9, then
so was the original number. So 819 is a multiple of 9, since 8 + 1
+ 9 = 18, but 777 is not, since 7 + 7 + 7 = 21.
• You can tell whether a number is a multiple of 4 just by looking at
its last two digits. If they are a multiple of 4, then so is the whole
thing. So 116 is a multiple of 4, just because 16 is. Similarly 5422 is
not a multiple of 4, as 22 isn’t.
• The number 8 is a little awkward, and there are various possible
ways forward. One is a variation on the test for divisibility by 4.
(Another is to give up and use a calculator!) If the last three digits
of the number are divisible by 8, then so is the original number. So
6160 is divisible by 8, since 160 is. The trouble is that telling
whether a three-digit number is divisible by 8 is not something
most people can do on sight. The best option is to divide the threedigit number by 2, and then apply the test for divisibility by 4. So if
we want to know whether 7476 is divisible by 8, first take the last
three digits (476) and then divide by 2 (238) and finally look at the
last two digits of that (38). In this case that is not a multiple of 4,
so the number fails the test.
• The fiddliest single-digit number is 7. There is a workable test
though, and it goes like this. To test 399 for divisibility by 7, chop
off the last digit (9) and double it (18). Then subtract that from the
truncated number (39 − 18 = 21). If the result is divisible by 7,
then so is the original number, which in this case it is. With this
test we might end up with 0: for instance if we apply the test to
147, we get 14 − 14 = 0. In this situation, 0 does count as a
multiple of 7, and so the number passes the test.
• The number 11 has a lovely test! It goes like this. Go through the
digits, alternating between adding and subtracting. If the result is
divisible by 11, then so is the original number. To test 9158, we go
9 − 1 + 5 − 8 = 5, which is not divisible by 11, so the test is
failed. It’s possible to end up with 0 again, or even negative
numbers, but that’s no problem. We do count 0 and −11, and
−22, and so on, as multiples of 11. So 1914 is a multiple of 11
since 1 − 9 + 1 − 4 = −11 is divisible by 11.
Breaking a number down into primes
Earlier in the chapter, we said that every number can be broken down
into primes, and we saw some examples. But if we are given a larger
number, such as 308, how can we actually find out what its prime
ingredients are? The idea is to try dividing by prime numbers in turn,
using the tests we’ve just seen. To start with, 308 is undoubtedly even.
So we can divide it by 2, this leaves 154. This is also even, so we can
divide it by 2 again, to get 77. Now, this is no longer even, so we
exhausted the 2s, and we move on to the next prime. We might try
dividing 77 by 3, but it fails that test. It is also easy to see that 77 is not
divisible by 5. So the next prime on the list is 7, and 77 is indeed
divisible by 7. Dividing it by 7 leaves 11, which is itself prime. So we
have finished. Collecting together all the primes that we divided by, we
get: 308 = 2 × 2 × 7 × 11.
The mysteries of the primes
The prime numbers are as mysterious as they are important, even today.
If you look at the sequence of prime numbers, there seems to be very
little order to it. Sometimes primes come very close together, like 11 and
13, and sometimes there are larger gaps such as between 199 and 211.
There are lots of seemingly basic facts about the prime numbers that we
still do not know for sure. One of these is Goldbach’s conjecture. In 1742,
Christian Goldbach noticed that every even number from 4 onward is
actually the sum of two prime numbers. So 4 = 2 + 2, 6 = 3 + 3, 8 =
3 + 5, … If you can prove that Goldbach’s conjecture must be true for
every even number, then you will have outshone the mathematicians of
the last two centuries. Although it has been verified up to an enormous
limit (around 1018—see The power of 10 for what this means), no-one
has yet managed to prove that it must be true for all even numbers.
Sum up To get to know a number well, you need to know which
other numbers divide into it. The most important ones to
check are the atoms of the mathematical world, the primes!
1 Break these numbers down into primes.
a 15
b 18
c 21
d 24
e 32
2 Which are true and which are false?
a “18 is a multiple of 3”
“18 is a factor of 3”
b “246 is a multiple of 5”
“5 is a factor of 246”
c “4 is a multiple of 108”
“108 is a factor of 4”
d “114 is a multiple of 6”
“6 is a factor of 114”
e “245 is a multiple of 7”
“7 is a factor of 245”
3 Test these numbers for divisibility up to 11.
a 64
b 42
c 75
d 176
e 68
4 Break these numbers down into primes.
a 30
b 210
c 108
d 189
e 1617
5 Goldbach’s conjecture! Write these even numbers as two primes
added together.
a 10
b 12
c 14
d 16
e 18
Negative numbers and the number line
• Understanding what negative numbers mean
• Recognizing when negative numbers are useful
• Knowing how to use the “number line”
If the idea of negative numbers does not come naturally to you, don’t
worry. You are in good company! It took mathematicians and scientists
thousands of years before the concept became respectable. But if you
don’t have a thousand years to spare, you needn’t worry either. The
principle is quite simple, once the basic idea has been grasped.
Negative numbers
The story of negative numbers begins in the world of commerce, and
they still demonstrate their great usefulness in trade today.
Imagine that I have set up a business, and am looking back over my
accounts at the end of my first month’s trade. There are three basic
positions that I might be in. Firstly, if my bank account is overdrawn,
that means that I am in debt. Over the month, I have spent more money
than I have received. So, how much money do I actually have, at this
stage? The true answer is “less than zero.”
The second possibility is that I have broken even. If my expenditure and
income have balanced each other out exactly, then the amount of money
in my account is zero. I am neither in debt, nor in credit.
The third possibility is that more money has come in than I have spent.
In other words, I have made a profit, and my bank account is in credit.
(Of course, this is a simplification from a business perspective, where
people generally distinguish between capital investment at the start of a
business, and running expenses. Nevertheless, the essential idea is, I
hope, reasonable enough.) In the past, people considered these three
separate possibilities as being essentially different. But, over time, the
realization dawned that the three could all be represented as different
positions along a single scale. Nowadays, we call this picture the number
The number line
The number line is a horizontal line, with 0 in the middle. To the right
of 0, the positive numbers line up in ascending order: 1, 2, 3, 4, … To
the left of zero are the negative numbers, which progress leftwards: −1,
−2, −3, −4, …
Sometimes is it convenient to put negative numbers inside brackets like
this: (−1), (−2), (−3), … There is nothing complicated going on here;
it is just to stop the—signs getting muddled up when we start having
other symbols around.
Notice that there is no −0. At least there is, but it is the same thing as
the ordinary zero: −0 = 0. Every other number is different from its
negative, so −1 ≠ 1, for instance.
We might think of this number line as representing my bank account. At
any moment, it is at some position along that line. If I am £15
overdrawn, I am at −15. If I am £20 in credit, I am at +20. (It is usual
to omit the plus sign, and just write “20,” but sometimes it is useful to
include it for emphasis.)
For this chapter, we will be focusing on the whole numbers (positive,
negative and 0). But between these are all the usual decimals and
fractions, which also come in both positive and negative varieties. We
shall meet these in more detail in future chapters. But if we want to find
, it is
of the way from 4 to 5. In the same way,
of the way
from −4 to −5. (A possible mistake here is to position it as
of the way
from −5 to −4.)
Negative numbers in the real world
For years, the principal purpose of numbers has been to count things: 3
apples, 7 children or 10 miles. So, when negative numbers first make
their entrance, a natural question is: how can you have −3 apples? I
hope that an answer is now plausible: having −3 apples means being in
debt by 3 apples.
How does the number line tie in with the usual idea of addition? Well
suppose you now go and pick 3 apples. But, instead of adding them to
your apple larder, you pay them to the person to whom you owe 3
apples. So, after receiving 3 apples, you end up with none: −3 + 3 = 0.
This can be shown on the number line as starting at −3, then moving
three places to the right to end up at 0.
It is not just trade where negative numbers are useful. Another example
is temperature. In the Celsius (or centigrade) scale, 0 is defined to be the
freezing point of water. If we start at 0 degrees and gain heat, we move
up into the warmer, positive temperatures. If we lose heat, we move
downward into the colder, negative numbers. A thermometer, then, is
nothing more than a number line, with a tube of mercury giving our
current position on it.
Moving along the number line
The number line is useful for seeing addition and subtraction at work. If
I am at 7, then adding 3 is the same as taking three steps to the right
along the number line: 7 + 3 = 10. Similarly, subtracting 3 is the same
as taking 3 steps to the left: 7 − 3 = 4.
This is not exactly news. But the same principle remains true whatever
the starting position. So even if we begin at a negative number such as
−5, then adding 3 again means taking three steps right: −5 + 3 = −2.
Similarly, subtracting 3 means taking 3 steps left: −5 − 3 = −8.
Negative negatives
Above, we saw how to use the number line to add or subtract. But there
is still something we need to make sense of: what is the relationship
between subtraction and negative numbers? In a sense they are the same
thing … but we need to know the details.
The trouble is that we seem to be using the same symbol (−) for two
different things: firstly (as in “−3”), this symbol indicates a position on
the number line to the left of 0, meaning a negative number; and
secondly, to describe a way of combining two numbers, as in “7 − 4.”
This second use corresponds to a movement leftwards along the number
So what is going on here? You could think of putting a minus sign in
front of a number as like “flipping it over,” using 0 as a pivot. So putting
a minus sign in front of 7 means 7 flips over, all the way to the far side
of 0, and lands on −7. So what then is “− −7,” or “−(−7),” as we
might write it? Well, when you flip over −7 you get back to 7. So: −
(−7) = 7
The fact that two minus signs cancel each other out in this way is the key
to working with negative numbers.
So when we face questions like “9 − (−3),” the two minus signs cancel
out, to give us “9 + 3.” But when we have “9 + (−3),” there is only
one minus sign, so it doesn’t get canceled out, and is the same thing as
“9 − 3.” This then is the relationship between negative numbers and
subtraction: Subtracting 3 from 9 is the same thing as adding −3 to 9.
Notice that this is not the same as adding −9 to +3.
So much for addition and subtraction. What about multiplying negative
numbers? Well to start with, remember that multiplication is essentially
repeated addition. So 4 × (−2) should be the same as (−2) + (−2) +
(−2) + (−2), which is just −2 − 2 − 2 − 2, that is to say −8.
To think about this in terms of trade, if I lose £2 each day (that is to say,
if I “make −£2”), then after four days I have lost £8 (or “made −£8”).
This illustrates that when we multiply a positive number by a negative
number, the answer is negative. So −5 × 2 = −10 and also 5 × −2 =
−10. Similarly, −1 × 4 = −4 and 1 × −4 = −4, and so on.
The most confusing moment in the dealing with negative numbers is
when two negative numbers are multiplied together: (−4) × (−2), for
example. But we have already seen above how two minus signs cancel
each other out, and here it is exactly the same again. Two negative
numbers produce a positive result: (−4) × (−2) = 8.
How does this work in terms of trade? Suppose I lose £2 per day (that is
to say, I “make—£2”). The question is how much will I have made or
lost in −4 days time? Well, “in −4 days time” must mean 4 days ago.
And if I have been losing money at a rate of £2 per day, then 4 days ago
I must have been £8 richer than I am today, which matches the result
We can put these rules in a little table:
Or more concisely:
When you have mastered negative multiplication, division is easy! All
we need to do is “multiplication backward.” So to calculate (−6) ÷
(−3), we have to solve (−3) ×
= −6. The two obvious possibilities
are 2 and −2, but only one can be right, so which is it? Well we know
that (−3) × (−2) = 6, which is not what we want. But (−3) × 2 =
−6, exactly as we might hope. So the answer is 2.
Perhaps surprisingly, the rules for working out the sign for division are
the same as for multiplication:
Or more concisely:
Sum up A number line is a great picture of the world of numbers:
positive, negative, and zero.
1 Draw a number line, between −5 and 15. Mark all the whole
numbers. Then add in marks for these numbers.
a − and
b −1 and −1
d −3 and
e −4 and 4
2 Add and subtract on a number line.
a 8 + 7 and 8 − 7
b 3 + 3 and 3 − 3
c 3 + 6 and 3 − 6
d −5 + 4 and −5 −4
e −2 + 3 and −2 −3
3 Doubling back a 5 + (−4) and 5 − (−4)
b 2 + (−3) and 2 − (−3)
c 0 − 5 and 0 − (−5)
d −4 −2 and −4 −(−2)
e −3 −5 and −3 − (−5)
4 Times tables go negative a 2 × (−3) and (−2) × (−3)
b 4 × (−5) and (−4) × (−5)
c 7 × (−3) and (−7) × (−3)
d 8 × (−4) and (−8) × (−4)
e 25 × (−4) and (−25) × (−4)
5 Division goes negative a 8 ÷ 2 and 8 ÷ 4 (−2)
b (−18) ÷ 6 and (−18) ÷ (−6)
c 28 ÷ 7 and 28 ÷ (−7)
d (−33) ÷ 3 and (−33) ÷ (−3)
e (−57) ÷ 19 and (−57) ÷ (−19)
• Interpreting decimals such as 0.0789
• Understanding what happens to the decimal point during arithmetic
• Mastering rounding
Not everything can be measured as whole numbers. It may not take a
whole number of minutes to walk to the shop, a recipe may not require
exactly a whole number of liters of milk. When we need to divide things
up more finely than the whole numbers allow, there are two main
approaches: fractions and decimals.
Neither method is better than the other; both are in use all the time. So
it is important to be able to translate between the two. As a very rough
rule of thumb, when the fraction is a simple one, it is best to use that: so
we might speak of “half an apple,” or “three quarters of a mile.” But it is
not practical to talk about “thirteen twenty-sevenths of a liter.” So when
real precision is needed, I recommend decimals.
Decimals—what’s the point?
With that all said, what exactly is a “decimal”? The idea comes from the
column representations of whole numbers that we have met in earlier
chapters. There, we had columns for units, tens, hundreds, thousands,
and so on, like this:
To incorporate things smaller than units, this system gets extended. We
introduce new columns for tenths of a unit, and similarly for hundredths,
thousandths, and so on, like this:
There is a mental adjustment we need to make when working with
decimals. When we are just writing whole numbers, we know that the
units column is the always the one furthest to the right. So “28” must
mean this:
But when we move into the realm of decimals, we have new columns to
the right of the units. Now the digits alone do not make it clear where
the units are, or which column is on the right. So “287” might mean:
or many other variations on the theme. This is a disaster!
The problem is solved with a new ingredient: the “decimal point.” This
is a dot which sits to the right of the units column. It is this point which
anchors the columns, allowing us to tell which is which. So “28.7”
while “2.87” means:
One consequence is that whole numbers directly translate into decimals
with the addition of 0s in the columns for tenths, hundreds, thousandths,
etc. So “34.0,” “34.00,” “34.000,” and so on, all mean exactly the same
as “34.”
(Some countries and languages use a “decimal comma” instead of a
decimal point, but it serves exactly the same purpose.)
Decimal arithmetic: addition and subtraction
The great thing about decimals is that the old column-based methods of
arithmetic transfer straight over to this new context. So to calculate 3.3
+ 5.8, we set it up in columns exactly as we did for addition of whole
numbers, just making sure to line up the decimal points of all the
numbers involved:
We proceed as before, making sure to start with the rightmost column,
and then carrying 1 to the next column as needed:
(If you’re not yet fully comfortable adding whole numbers, you might
want to revisit the chapter on addition.) What goes for addition is
equally true of subtraction. To calculate 6.2 − 2.4, we set it up in
columns like this:
Again the method is identical to that for whole numbers, presented in
the chapter on subtraction. So we begin with the rightmost column, and
borrow Is as necessary:
Multiplication: moving rightward
Like addition and subtraction, multiplcation also translates easily to the
decimal context. The main thing to watch out for is the position of the
decimal point, or more accurately the positions of the digits relative to
the decimal point. It is a good habit to think of the decimal point as
being fixed and immovable, while the digits around it shuffle leftwards
or rightward.
To see what we mean by the “position of the digits relative to the
decimal point,” let’s look at the calculation 3 × 2, but with these digits
in different places relative to the decimal point. Let’s start with 3.0 ×
2.0. There are no surprises here, since this is nothing more than 3 × 2,
which we know to be 6. Another easy one is 3.0 × 20.0 = 60, which we
might calculate as 20 + 20 + 20 = 60. Alternatively, we could first
work out 3 × 2, and then shunt the answer one column to the left,
filling in the empty column with a zero, again giving 60. (This is the
method presented in the chapter on multiplication.) Next, what is 3.0 ×
0.2? The answer is 0.6. We can see this easily, because 0.2 + 0.2 + 0.2
= 0.6.
But how does this fit in with the column depiction of the calculation?
The golden rule here is this: just as multiplying by 10 moves the digits
one step to the left, so multiplying by 0.1 moves them one step to the
right. Why should this be? One answer is that multiplying by 0.1 (that is
to say
) is the same thing as dividing by 10.
To calculate 3 × 20, the first step was to realize that 3 × 2 = 6, and
the second was to shift everything one column to the left, filling in any
empty columns with zeros, to arrive at 60. Calculating 3 × 0.2 is almost
the same: first we calculate 3 × 2, and then shunt everything one
column to the right, again filling any empty columns with zeros, to arrive
at 0.6.
The same line of thought works when we look at 0.3 × 0.2. As before
we first work out 0.3 × 2, which is 0.6, and then shunt everything one
step to the right, to arrive at an answer of 0.06.
Multiplication: putting it together
With the golden rule in place, more complex multiplication can be put
together just as for whole numbers. The pattern follows the column
method for whole number multiplication exactly (so make sure you are
comfortable with that before proceeding with this). To calculate 21.3 ×
3.2, we would set it out like this:
We start on the right, by multiplying the whole top row by the rightmost
digit on the bottom row. In this case, that means multiplying it by 2, and
then moving everything one step to the right:
Then we move to the next digit of the second number, in this case 3, and
again multiply the top number by that. This time no shunting, left or
right, is required because 3 is in the units column:
Finally, we add up the two numbers below the line to arrive at the final
Notice how the decimal points are kept in line throughout the
calculation. This is good practice, as it means all the columns marry up
We use decimals when we want a more accurate description than whole
numbers can provide. But we may not always want the laser-sharp
accuracy that decimals of unlimited length provide. This might sound
strange, but think of it this way: do you need to know the quantity of
butter to use in a cake, down to millionths of a gram? It is useful for me
to know my weight to the nearest kilogram, or even in exceptional
circumstances to the nearest gram, but never to the nearest nanogram
(that is, 0.000000001g, or one billionth of a gram).
Very often we want to round decimals to some chosen level of accuracy.
If I ask your height to the nearest centimeter, you might give me an
answer of 1.64 meters. In doing so, you have rounded your answer to
two decimal places.
Rounding is something we shall use throughout this book. So how does
it work? We begin by specifying a level of accuracy, usually as a number
of decimal places. The decimal places are the columns to the right of the
decimal point: those representing tenths, hundredths, thousandths, etc.
Suppose we decide on one decimal place as a suitable level of accuracy
for measuring the weight of a helping of dog-food. This means we will
want to round our answer to the nearest tenth of a kilogram. The scale
reads 3.734kg. So, when we round it, the answer is 3.7kg.
It is tempting to think that rounding is simply a matter of chopping the
number off at a suitable point. Actually there is a little more to it than
this—but just a little. For example, suppose we want to round 3.699 to 1
decimal place. Chopping the end off would give 3.6, but in fact 3.699 is
much closer to 3.7. It is just 0.01 away from 3.7, whereas it is 0.099
away from 3.6. So 3.7 is the correct answer here.
Now, what if we wanted to round 5.46 to 1 decimal place? The two
candidates are 5.4 and 5.5. But 5.46 is 0.06 away from 5.4, and just 0.04
away from 5.5, so 5.5 is the right answer. On the other hand, if we start
with 2.13, then the two candidates are 2.1 (0.03 away) and 2.2 (0.07
away). So, this time, the answer is 2.1. Thinking about different
examples like these produces the following rule: • To round a number to
a certain number decimal places, chop it off after those digits.
• Then look at the first digit of the tail (the part that was chopped
• If that digit is between 0 and 4, then leave the answer as it stands:
just the original number, truncated.
• But if the first digit of the tail is between 5 and 9, then the final
digit of the truncated number should be increased by 1.
This rule is, admittedly, a bit of a mouthful. But the procedure itself is
not really very difficult, with a little practice. The trickiest case is when
we have an example like 19.981, to be rounded to 1 decimal place. We
begin by chopping off after 19.9. Then, the first digit of the tail is 8. So
we know we have to increase 19.9 by 0.1 to 20.0. In this case the
rounded answer doesn’t actually look anything like the original number,
which might be confusing at first.
Incidentally, when rounding to one decimal place, it is good practice
always to include that decimal place, even if it contains a zero. So in the
above example, we would leave the answer as “20.0” rather than
abbreviating it to “20.” The reason for this is that “20.0” communicates
the level of accuracy to which you are working, namely one decimal
There will be plenty more practice in rounding later in the book.
Sum up If you can do arithmetic with whole numbers in
columns, then it is only a small step to extend the technique to
1 Set out in columns and add a 3.2 + 2.3
b 6.4 + 6.7
c 12.31 + 3.19
d 6.78 + 3.33
e 0.00608 + 0.00503
2 Decimal subtraction (columns again) a 7.9 − 3.6
b 6.43 − 2.31
c 9.6 − 1.7
d 7.67 − 3.48
e 19.72 − 9.89
3 Digits move right a 0.2 × 4
b 0.2 × 0.4
c 0.5 × 7
d 0.5 × 0.7
e 0.5 × 0.07
4 Full-blown multiplication a 2.3 × 1.7
b 6.2 × 5.2
c 3.4 × 2.9
d 25.7 × 6.8
e 5.72 × 7.9
5 Round these.
a 5.3497 to one decimal place
b 0.16408 to two decimal places
c 0.16408 to one decimal place
d 9.981123 to one decimal place
e 0.719601 to three decimal places
• Interpreting fractions
• Recognizing when two fractions are the same
• Understanding top-heavy fractions
• Translating between fractions and decimals
Working with fractions ought to be easy. All it involves is cutting up
cakes into suitably sized slices, and then counting the pieces. But
somehow this simple idea can turn into a nightmare of “common
denominators” and “lowest common multiples.”
There is one key thing to understand to be able to work with fractions. It
is that any fraction, such as , can be rewritten in different ways, for
example as
. The crucial question is this: how can we tell when
two fractions are really the same? Might
also the same thing as
, for
An old reliable method is to start with two identical cakes and divide up
each according to the two fractions we are thinking about. If the two
resulting helpings are the same, then so are the two fractions. So, if we
divide one strawberry pavlova into eighths, for example, and serve up
four of them, and divide another pavlova into two halves, and serve up
one of them, then the two helpings are indeed the same. So,
. But if
we divide a cake into twenty-sevenths, and dish out 13 of them, we do
not quite get a helping equal to half a cake. So
What we need is a way to get this information directly from the
numbers, without having to bake any cakes. As it happens, it is not too
hard in this particular case: four eighths is the same as one half, because
4 is half of 8. On the other hand, 13 is not half of 27. This is fine, but
some numbers are less easy to manipulate than . What this is hinting
toward is the golden rule which tells us how to change a fraction’s
appearance, while keeping its value the same.
So, for example,
is true, because the second fraction comes from
multiplying the top and bottom of the first fraction by 4. A less obvious
example is that
, as we can see by multiplying the top and bottom
of the first fraction by 6. In terms of cakes, the golden rule tells us how
to slice the helping of cake into thinner pieces, while keeping the overall
helping the same size.
Simplifying fractions
Here is some jargon: the top of a fraction is known as the numerator, the
bottom is called the denominator. The golden rule says that if you have a
fraction then multiplying the numerator and denominator by the same
number may change the superficial appearance of the fraction, but not
its underlying value. Now, this rule also works backward: if instead you
divide the top and bottom by the same number, then the fraction’s value
is similarly unaffected.
So if we start with a fraction such as
, and then we divide both the top
and bottom by 5, we see the fraction in a new form: . The useful thing
about this is that the numbers on top and bottom have become smaller.
The fraction therefore seems simpler. This process of dividing top and
bottom by the same number is known as simplifying the fraction.
A nice thing about
is that there is no number which can divide both
the top and the bottom: this version of the fraction is as simple as it gets.
We might say that it has been fully simplified.
When writing a fraction it is usually good practice to present it in fully
simplified form. This means checking whether there is any number
which divides both the top and bottom, and if there is, then dividing top
and bottom by it. So starting with
, we might notice that the top and
bottom are both divisible by 2. Dividing top and bottom produces ,
which is now in fully simplified form.
The ultimate in simplification is when we have a fraction such as
Here the top and the bottom are both divisible by 4. Dividing top and
bottom gives . But what is
Well, it is 3 ÷ 1, which is 3. Whenever
there is only a 1 on the bottom of a fraction, the bottom effectively
disappears, leaving just the top on its own as a whole-number answer.
(Going the other way, if we want to write 5 as a fraction instead of a
whole number, we can just put a 1 underneath it, to get . The golden
rule then tells us that this is the same as
, and other variations on
the theme. Notice that this matches our intuition, as 10 ÷ 2 and 15 ÷ 3
are indeed equal to 5.)
Top-heavy fractions
Most of the fractions we have seen so far in this chapter have had a
smaller number on top of larger one. But there is no law that says it
must be so:
is a perfectly valid fraction too, and everything that we
have said so far applies equally to “top-heavy” fractions like this (as does
everything we say in the next chapter).
Most scientists don’t bat an eyelid at fractions like . But in the wider
world they can seem a bit strange. Instead of saying “three halves of an
hour,” most people would say “one hour and a half,” which might be
written like this: 1 .
Expressions like
are accurately described as top-heavy, or, somewhat
unfairly, as improper fractions. An expression like 1 is called a mixed
number. Notice that there is an invisible “+” sign here: 1 is really the
same thing as
So much for the jargon. The point is that 1 and
are actually the same
thing. They are just presented in slightly different ways. The next
challenge, then, is to be able to translate between the languages of
improper fractions and mixed numbers.
Suppose we are given a top-heavy fraction such as
and we want to turn
this into a mixed number. The process is not too tricky: in fact it comes
directly from the meaning of a fraction. Don’t forget that a fraction
represents division. So
is exactly the same thing as 7 ÷ 3. Having said
that, let’s calculate 7 ÷ 3. The answer is that 3 goes into 7 two times,
leaving a remainder of 1. The 2 is the whole number part of the answer,
and the remainder 1 then goes on top of the fractional part, while 3 goes
on the bottom, giving
. (We saw this method in the chapter on
division.) The general rule is:
To express a top-heavy fraction (such as ) as a mixed number, the
number of times the bottom fits into the top (2) goes outside the fraction as
a whole number. The remainder (1) then stays on top of the fraction, and
the bottom of the fraction (3) doesn’t change:
Going the other way is easier:
If we are given a mixed number (such as
) that we want to change into
an improper fraction, then we start by multiplying the whole number (2) by
the bottom of the fraction (4), which gives us 8. Then we add on the top of
the old fraction (3), to get the top of the new one (11), and the bottom (4)
stays the same. So the answer is
From decimals to fractions
We have met two ways of expressing non-whole numbers: fractions and
decimals. Since both of these languages are very common, it is important
to know how to translate between the two.
The simplest examples are worth memorizing:
Beyond these, some techniques are required. Suppose we are given a
decimal 0.4. How can we express this as a fraction? Actually, decimals
are fractions already. Why is that? Well, remember that the columns to
the right of the decimal point represent tenths, hundredths, thousandths,
and so on (just as the columns to the left represent units, tens, hundreds,
etc.). So 0.4 really is just
. All that remains is to simplify it, which we
can do by dividing the top and bottom by 2, to give a final answer of .
Let’s take another example: 0.35. We know 0.35 is three tenths plus five
hundredths. Or, to put it another way, it is thirty-five hundredths:
We can simplify this by dividing the top and bottom by 5, to get
In the last example, 0.35 had two decimal places, so we had to express it
in hundredths initially, before simplifying. Similarly, if we want to
convert 0.375 to fractional form, we need to express it first as
. The we can simplify by dividing the top and bottom by
5 three times, to get .
From fractions to decimals
Suppose we are given a fraction and want to express it as a decimal.
First let’s take a nice friendly one, , and pretend that we don’t already
know what the decimal equivalent is. How might we work it out?
In the last section, before we simplified the result, the fractions we
arrived at looked like this
, and so on. If we can convert
a fraction of one of these types, then we will nearly be there. So we need
to apply this chapter’s golden rule. We will try these possibilities in turn.
The bottom of our fraction is currently 4. But we want to change it to
10, or 100, or 1000, etc. Unfortunately, there is no whole number we can
multiply 4 by to get 10. So we cannot express
as tenths. Moving on,
however, there is a number we can multiply 4 by to get 100, namely 25.
So applying the golden rule, we multiply the top and bottom of
to get
by 25
. It is now easy to recognize that this is the decimal 0.75.
Recurring decimals
Now, we run into an awkward fact. Some fractions have decimal
representations which look a bit strange. Start with . If you type
into your calculator, you should get an answer of
0.3333333333. In fact this is not an exact answer; the string of 3s really
goes on forever. Try decimal short division for 1 ÷ 3 and see what
This is what is known as a recurring decimal, meaning that it gets stuck in
a repeating pattern that goes on forever. A lot of fractions do this. For
When calculating with these sorts of numbers, there is no choice but to
round them off, after a number of decimal places, as the calculator does.
However, there is a special notation for recurring decimals: a dot over
the repeating number. So we would write
Some numbers have more complex repeating patterns. For instance
, which we would express as
Every fraction will have an expression as either a terminating decimal
(such as 0.51) or a recurring decimal (such as 0.51). There are other
numbers where the decimal expansion goes on forever without ever
repeating; these are the so-called irrational numbers. Famous examples
are π (see Circles) and
(see Pythagoras’ theorem).
Sum up Once you have got to know them, fractions really are a piece of
1 Different yet the same. Write
a Twelfths
b Eighths
c Sixteenths
d Twentieths
e Hundredths
2 Simplify
3 Top-heavy fractions to mixed numbers
4 Decimals to fractions
a 0.9
b 0.6
c 0.95
d 0.625
e 0.875
5 Fractions to decimals (possibly recurring)
6 Which numbers on the bottom of fractions produce recurring
decimal? Experiment and see!
Arithmetic with fractions
• Knowing how to add and subtract fractions
• Canceling fractions, to speed up multiplication
• Understanding how to divide fractions
In the last chapter we saw how to represent numbers as fractions,
noticing particularly that one number has many different fractional
representations. In this chapter we will see how to do arithmetic with
fractions: adding, subtracting, multiplying and dividing them.
Surprisingly, it is addition and subtraction which are trickier in this
context: multiplication and division are fairly straightforward. A
common error is to see
and add the top and bottoms separately to
get . This is certainly wrong (think of half a cake being added to
another three quarters of a cake). So let’s jump in at the deep end, and
get started adding fractions the correct way.
Adding fractions
Everyone can agree that adding fractions is easy, in some circumstances
at least. What is one ninth plus one ninth? Two ninths, of course. What
It must be
. As long as you can add ordinary whole numbers,
you can add these sorts of fractions. What makes these so simple is that
these fractions all have identical bottom numbers. (In the jargon, they
have a “common denominator.”)
Changing the bottom numbers
The key to adding all fractions is to be able to transform any addition
into an easy one like those we have just looked at. Take a simple
example: what is
Slicing up a cake quickly reveals the answer: .
But we want a calorie-controlled way to see this directly from the
numbers, skipping the cakes.
The crucial observation is that the half can be broken down into two
quarters, using the previous chapter’s golden rule for fractions. Now we
can rewrite the question. Instead of
we can write it as
. This
brings us back to the familiar ground of fractions with the same bottom
number, which we can then add straightforwardly. From this we get the
golden rule for this chapter.
In the last example, we were able to split a half into two quarters. A
similar trick will work with lots of other fractions. Suppose we want to
. This time we can break the single sixth into two twelfths
(using the previous chapter’s golden rule), and so change the question to
. Once again we are back in the comfortable realm of matching
bottoms! Another example is
. This time, we split the half into four
When cakes collide
A thornier type of problem is one like this:
. The trouble with this is
that we cannot split a half into thirds, or a third into halves. Neither fits
into the other. This time, to get to a sum in which the two fractions have
matching bottom numbers we are going to have to alter both fractions.
But which numbers should we multiply their tops and bottoms by? Let’s
try some out. If we start with , applying the previous chapter’s golden
rule with 1, 2, 3, 4, 5 in turn produces the fractions
equal to . Meanwhile, starting with
, etc., all
, etc. Looking
at these two sequences, we might spot that they both contain a fraction
with 6 on the bottom. This is promising, as it allows us to rewrite
, and then proceed as before.
Why did the number 6 work so well here? Looking at the bottom row of
the sequence
, what we see is nothing other than the two times
table. Similarly the bottom row of
is the three times table. The
number 6 worked perfectly then because it features in both times tables.
(In fact it is the first number to feature in both; this is the so-called
“lowest common multiple.”)
So, when faced with the sum of fractions,
, the first thing to do is to
find a number which features in both the three and five times tables.
Thinking about it, 15 is such a number. Then we can apply the golden
rule twice, to turn
into a sum of two fractions with 15 on their
bottoms. To get 15 on the bottom of , we must multiply by 5. So,
according to the golden rule, we must multiply the top by 5 too:
Similarly, to get 15 on the bottom of , we have to multiply top and
bottom by 3. This produces
can easily finish off to get
, and our sum becomes
, which we
The final step, when adding fractions, is to make sure that the answer is
fully simplified, since this doesn’t happen automatically. For instance if
we add
, by following the rules above, we can turn this into
which then becomes . But
can be simplified to give , which is the
final answer.
Subtracting fractions
The rules for subtracting fractions are almost identical to those for
adding them. The only difference is that, where previously we added,
now we subtract. Genius! So, to take an example, if we have
exactly as before we change this to
, then
. We can then evaluate it as .
Since this can’t be simplified, we have finished.
If you are comfortable with adding fractions, then subtraction should be
a piece of cake.
Multiplying fractions
While adding and subtracting fractions is a slightly tricky procedure,
multiplying them is straightforward. For instance, to calculate
we do is multiply the numbers on the top (2 × 1), and the numbers on
the bottom (3 × 5), and then put the answer back together:
The procedure may be easy, but what is actually going on, in the
language of slices of cake? Well, suppose you have sliced a cake into
fifths. Calculating
amounting to
would correspond to a serving of two such slices,
of a cake. Similarly, the calculation
a serving which comprises
amounts to
gives the size of
of one those slices. As we have seen, this
of the whole cake. This is a case of multiplication being
implied by one short English word: “of.” What
really means is
Of course, after multiplying fractions you may need to simplify the
answer (as you also need to do when adding or subtracting them). In this
case, though, a shortcut is sometimes possible.
To see how, let’s take another example:
. The rules for multiplying
fractions (together with knowledge of times tables) make it simple to
come up with an answer:
. The final step is to simplify this fraction. So
the question is: is there any number which can divide both the top
number and the bottom number? A little reflection will reveal an
answer: 7. So dividing top and bottom by 7 produces the final result: .
That all worked perfectly well, but going back to the original question
we can actually see the final answer
immediately if we know
how to look for it, without having to work through all the multiplication
and simplifying.
The question amounts to this:
. On the bottom of this fraction, even if
we have forgotten all our times tables, we can see straight away that this
number is divisible by 7. And the same goes for the top. So we can move
directly to simplifying, by dividing out by 7 immediately, which we
might write like this:
This process is often known as canceling, and it is a great labor-saving
device. It comes into its own when multiplying more than two fractions
together. Suppose we are faced with
. We can immediately set
to canceling the 3 on the top with that on the bottom, the 7 on top with
the one below, and similarly for the 5S:
. This can finally be
simplified to . Once you get the gist of this, it is much easier than
evaluating it as
, and then fiddling around trying to simplify it from
As another example, take
. This time, the numbers on the top and
bottom are all different. Yet some canceling is possible nevertheless. The
15 on the bottom is divisible by 5, which occurs on the top. So we can
cancel those out, leaving 3 on the bottom (because 3 × 5 = 15).
Similarly, we can cancel the 7 on the top with the 28 on the bottom, to
leave 4 on the bottom
When everything is canceled from one line, as happens on the top in this
case, there is always an invisible 1 which remains. So the final answer in
this case is
Dividing fractions
What does it mean to ask what 10 ÷ 5 is? I’m not interested in the
answer (that’s easy), but rather the meaning of the question. Well, the
answer should be the number which tells us how many times 5 fits
inside 10. To put it another way, the question we have to solve is 5 ×
= 10.
So far, so good. But what happens when we ask the same question of
fractions: what is
According to the same philosophy, it ought to be
the number of times that
. The answer to this is 3.
fits inside . To put it another way,
This is all perfectly correct, but it is a common source of confusion that
dividing one small number by another small number can produce
something much bigger. Shouldn’t division always produce smaller
results? The answer is a categorical “no”! Dividing by a large number
gives a smaller answer, but dividing by small numbers (meaning smaller
than 1) gives bigger answers.
To go back to 10 ÷ 5, it is no coincidence that the answer here is the
same as the answer to
as does
. Why? Because 10 ÷ 5 means the same as
What this suggests is the following rule for dividing fractions:
To divide one fraction by another, turn the second one upside down, and
then multiply.
So we would write the above example out as
Another example might be
division, this is equal to
. According to the rule for fractional
, which, after canceling, becomes .
A word of warning: sometimes fractional division can look very
different, when the division symbol is itself replaced with a fraction. For
instance, the last example might also be written as . Don’t be put off by
this—the meaning is exactly the same as before. And what if we saw
something like
same as
Well this has the same meaning as
, and so equal to
, which is the
Sum up Fractional arithmetic is a piece of cake, so long as you
remember the rules!
1 When bottoms match
2 Mismatching bottoms
3 Double trouble
4 Cancel down first!
5 Fractions turn upside-down
• Understanding what powers are and what they are useful for
• Understanding how bank interest works
• Understanding exponential growth
Ultimately, multiplication is repeated addition. So 5 × 4 is four 5s
added together: 5 + 5 + 5 + 5. In the same way, we can step up a
level and look at powers—the name for repeated multiplication. So 54
(that’s pronounced “5 to the power 4”) is four 5s multiplied together:
54 = 5 × 5 × 5 × 5.
What is the point of these powers? One answer to that question comes
from geometry, where they crop up all the time, as we shall see in the
chapter on area and volume. In fact, there are special names for certain
powers, which already suggest geometrical ideas.
Raising to the power 2 is known as “squaring,” so 52 or 5 × 5 is also
known as “five squared.” The reason for this is that if you draw a square
whose length is 5cm, then its area is 52cm2.
In the same way, raising to the power 3 is known as “cubing.” So 53 (or
5 × 5 × 5) is “five cubed.” Why? Because a cube with sides each 5cm
long has volume 53cm3 (see Area and volume).
Calculators and powers
Powers are usually written as superscripts: 512. If you are working on a
computer, however, the symbol Λ is often used instead: 5Λ12. As we
shall see shortly, powers grow very large very quickly, so using a
calculator or computer is often unavoidable. Most calculators will have a
button marked
(or perhaps
) with which to calculate powers. (If
this symbol appears above another button, you may need to press the
“SHIFT” or “Second Function” button first to access this function.) To
calculate 512, you would need to press
Sessa’s chessboard
Another word for a power is “exponent.” So in 53 the exponent is 3.
Exponential growth is famous for being extremely fast. In fact, there is a
myth about this, which concerns an Indian wise man called Sessa, said to
be the original inventor of chess. When he took his invention to show
the King, His Majesty fell in love with the game. In fact, he was so
impressed that he declared that he would give Sessa anything he wanted
as a reward, anything at all. Sessa’s reply was the stuff of legend.
Laying out his chessboard in front of him, he said that he would like 1
grain of wheat on the first square of the board. On the second square, he
would like 2 grains of wheat, and on the next square 4 grains, and then
8, and so on, continuing in the same manner until each square was full.
The King was annoyed that his generosity should be ridiculed, thinking
that a few grains of wheat was not much to request of a great leader. But
how many grains of wheat would the board require?
In quiz 3 you have to fill in the numbers of Sessa’s chessboard as far as
you can, without a calculator. If you can get as far as the 21st square,
you are doing well! A chessboard has 64 squares: if you can complete
them all, you are doing better than some pocket calculators.
Exponential growth
In the story of Sessa’s chessboard, there is 1 grain of wheat on the first
square. It is a mathematical convention that any number raised to the
power of 0 gives 1. So we can write this as 20. Don’t worry if this seems
a little strange; it is just a rule we adopt to make sense of otherwise
meaningless expressions like 20. It turns out to be convenient, as we
don’t have to keep talking about 0 separately, as a special case.
On the second square of the board, there were 2 grains, that is to say 21.
On the next, there were 4, that is 22, and on the next 8 = 23, and so on.
It is the power (or exponent) of 2 which grows by 1 from square to
square, making this a classic example of “exponential growth.” As the
story shows, exponential growth can be very fast indeed. The King
certainly had to give more than he bargained for: by the 51st square the
King would have to hand over the entire global wheat harvest for 2007.
Exponential decay
We have seen that powers can grow very quickly. But it is not always so.
It is certainly true that if we take a number such as 2, and multiply by
itself, then the answer is bigger. As Sessa knew, the more times we
multiply it by itself, the bigger it gets. But not all numbers are like this.
If we take a number like , then when we square it we get something
. If we cube it, we get something smaller still:
, and so on.
This is an example of exponential decay: the flipside of exponential
growth. As with exponential growth, the process is very quick indeed:
is less than a millionth.
Where is the boundary between exponential growth and exponential
decay? Which numbers grow big when repeatedly multiplied by
themselves, and which shrink away toward 0? The dividing line is the
number 1. When you raise 1 to a power, nothing happens as 1 × 1 × 1
× … × 1 is always 1, no matter how many multiplications there are.
But anything less than 1 (but still greater than 0) will decay, and
anything bigger than 1 will grow.
What is surprising is how quickly this happens, even for numbers which
are only just on one side or the other of 1. For example, 0.922 is already
less than 0.1, while 1.122 is more than 8.
The arithmetic of powers
For the final part of this chapter, let’s look a little deeper at the
arithmetic of powers. How do we work out 24 × 34? Writing this out,
we get:
So that’s four 2s multiplied by four 3s. But we know that when we
multiply things together the order doesn’t matter. So we can rearrange
this as:
This is the same as 6 × 6 × 6 × 6, or 64.
This is an example of a general rule that, for any numbers a, b, c, ac × bc
= (a × b)c
What happens when we multiply two powers of the same number
together? For example, what is 23 × 24? Writing out the powers as
multiplication, the answer is,
. Counting up the
number of 2s gives the answer: 27.
It is no coincidence that 23 × 24 = 27 and also 3 + 4 = 7. This is one
of the rules (or laws) for powers. It says that, for any numbers a, b, c, it
will always be true that: ab × ac= ab + c (The first law of powers) What
this says is that to multiply two powers together you add their exponents
(but this only works if they are powers of the same number!).
Another rule or law tells us what happens when we look at a power of a
power, for example: (23)4. What is this? Writing it out, we get, 23 × 23
× 23 × 23, and expanding these we get:
There are twelve 2s here altogether, because 3 × 4 = 12, so we get an
answer of 212. This gives us the general rule, for any numbers a, b, c,
(ab)c = ab × c (The second law of powers)
Sum up From exponential growth, to the laws of powers, powers
are subtle calculations. But remember, really there’s nothing
more to them than repeated multiplication!
1 The power of powers: write these out in full.
a 33
b 62
c 53
d 34
e 63
2 Even more powerful powers: calculate the values.
a 55
b 66
c 124
d 1252
e 895
3 Continue the sequence of numbers from Sessa’s chessboard.
1, 2, 4, 8, 16, …
4 Work these out leaving the answer as a fraction.
5 Use the arithmetic of powers to express each of these as a single
number to a single power (e.g. 816).
a 214 × 28
b (25)3
c 35 × 55
d 1319 × 137
e 2519 × 419
The power of 10
• Knowing the names of extremely large numbers
• Being able to write very large and very small numbers
• Understanding the metric system of measurements
There is something special about the number 10. The numbers 0–9 each
have their own individual symbol. But when we reach 10 its symbol is
made up from those for 0 and 1. This simple observation cuts to the
very heart of the modern way of representing numbers. It was not
always like this, as anyone familiar with Roman numerals knows.
It is not just 10 which is significant, but all the powers of 10. These are
100, 1000, 10,000, 100,000, and so on. These are special in our way of
writing numbers, since they mark the points where numbers become
longer: while 99 is two digits long, 100 is three, while 999 is three digits
long, 1000 is four, and so on.
The number 1000 is 10 × 10 × 10, which, in the language of powers is
103. (See the previous chapter for a general discussion of powers.) The
important observation is this: the power of 10, in this case 3, actually
counts the zeros. So 103 is the same as 1 followed by three zeros. This
becomes very useful as the numbers get larger. The expression 1010 can
be read and digested much more easily than if you were left to count the
zeros yourself: 10,000,000,000.
Powers of 10 are also the points where new names for numbers appear.
If we scroll down the powers of 10, the first few are simple enough:
After a thousand, new names appear every three steps, with ten and a
hundred filling the intermediate gaps. So:
It’s the multiples of three which are important from the point of view of
naming numbers.
A word of caution here: in the past there was some disagreement across
the Atlantic about what constituted a “billion.” Americans have always
considered a billion to be a thousand million (that is, 109) while the
British used to use the same word to mean a million million (that is,
1012). That is no longer the case. Today, the system above is universal in
the English-speaking world. However, the disparity is worth
remembering, if you are ever reading British documents dating from
before 1974. In other languages, systems vary. In French, for example,
106 is un million, 109 is un milliard, and 1012 is un billion. In Japanese,
Chinese and Korean, the basic unit is 104 rather than 103 with new
numerical names appearing at 104, 108, 1012, 1016, and so on.
Translators beware!
The metric system
Often when we see numbers written down, there are a few letters after
them: for example, 5kg, 10s, 12cm. These letters are different from the
ones that appear in algebra (see Algebra). Instead these are units, and
their purpose is to define exactly what the numbers are measuring,
whether that be mass, time, distance or something else, and the scale
being used to measure it.
There is a whole army of units that people use to measure everything
from humidity (g/m3, that is, grams per cubic meter) to the spiciness of
chilies (SHU, that is, Scoville heat units). It would not be practical or
useful to attempt a complete list!
Nevertheless, there is something important to say about the way that a
certain class of unit relates to the powers of 10 we have just been
looking at. This is known as the metric system, and it is based on meters
for distance (rather than inches or miles), grams for weight (rather than
ounces or stones), seconds for time (rather than minutes or years). Other
units are then built out of these. A liter, for example, is 100cm3 (see Area
and volume), while the standard measure of force is the Newton, defined
to be 1kg m/s2.
Let’s take an example, to see how the system works. A gram is a unit of
weight. One gram on its own is not very much. So if we want to measure
people, cars or planets, a gram doesn’t seem a very satisfactory starting
point. However, there are prefixes which can be put in front of the word
“gram,” to make the unit bigger. One is “kilo” which means a thousand.
So 1 kilogram is the same thing as 1000 grams. The kilogram is a
sensible unit for measuring the mass of a person, for example. Just as
with the names of numbers, new metric prefixes generally occur every
multiple of 3:
Many modern scientists might frown at the first two in this list, but
mechanical engineers do sometimes discuss force in terms of decanewtons
(daN), while meteorologists occasionally measure atmospheric pressure
in hectopascals (hPa). But it is for larger numbers that the system really
becomes useful. It is very common to measure distances in kilometers
(km), and the resolution of a digital camera might be 5 megapixels (MP),
meaning that it contains 5 million individual image sensors (pixels).
Unless you are an astrophysicist measuring the weight of stars, the
largest of these prefixes you will probably ever need is ‘tera-‘, or 1012. It
is quite common for computers to have 1 terabyte (TB) disk drives now.
If you wanted to measure the weight of a car, the megagram would be a
sensible unit to use. It just happens that the megagram more commonly
goes by the name of ton, meaning a million grams, or equivalently, a
thousand kilograms.
Small things
Everything we have said for the very large goes equally well for the very
small. We can use negative powers of 10 to represent small numbers. To
start with, 10-1 means
, or equivalently 0.1. Similarly, 10-2 is
, or 0.01, and 10-3 is
which is
. which
, or 0.001, and so on.
A quick rule, as before, is that the negative power counts the number of
zeros, with the important caveat that a single zero before the decimal
point must be included in the count. With this said, it is easy to see that
10-6 is one millionth, 10-9 is one billionth, and so on. Writing these in
decimals, we get 10-6 = 0.000001 and 10-9 = 0.000000001.
There are also metric prefixes for the small numbers:
Again the first of these is less commonly used than the second and third,
although the standard measure of loudness, the decibel, was originally
defined as one tenth of a bel (a unit which has long since fallen out of
favor). We commonly use a centimeter (cm), which is a hundredth of a
meter, and a milliliter (ml), which is one thousandth of a liter. A pill
might contain 5 micrograms µg) of vitamin D, and we have all heard the
hype surrounding nanotechnology, meaning engineering which takes
place on the scale of nanometers (nm). Getting very small, 3 picoseconds
(ps) is the time it takes a beam of light to travel 1 millimeter.
Standard form
Metric prefixes and funny names like “sextillion” are sometimes useful,
and are good fun. But actually, with powers of 10 at our disposal, they
are not strictly necessary.
Your calculator, for instance, can function perfectly well without them.
Type in two large numbers to be multiplied together, perhaps
20,000,000 × 80,000,000. How does your calculator display the result?
Mine displays “1.6 × 1015”. We could translate this as
1,600,000,000,000,000 or “1.6 quadrillion.” But actually the expression
“1.6 × 1015” is shorter and easier to understand than either of these. My
calculator is taking advantage of powers of 10, to express this very large
number in an efficient and compact way: 1.6 × 1015. This way of
representing numbers is known as standard form.
It is an essential skill to be able to move back and forth between
standard form and traditional decimal expressions, and that is the final
topic we shall explore in this chapter.
In technical terms, a number is in standard form if it looks like this: A ×
10B, where A is a number between 1 and 10 (not necessarily a whole
number), and B is a positive or negative whole number. An example is
3.13 × 104. Let’s translate this back into ordinary notation. Above, we
saw that the power of 10 can correspond to the number of zeros on the
end (so 103 = 1000 for instance). This is fine when we are considering 1
followed by a line of zeros, but 3.13 is not like this. We now need a
slightly more sophisticated perspective. The answer comes from the
chapter on multiplication, where we saw that multiplying by 10
corresponds to shifting the digits one step to the left with respect to the
decimal point. So multiplying by 104 is equivalent to shifting to the left
four times: 3.13 → 31.3 → 313.0 → 3130.0 → 31300.0
That final number, 31,300, is our answer. (This should not come as a
surprise, since 104 = 10,000 and 31,300 is 3.13 lots of 10,000.) Another
way to think of this is that the 104 tells us the length of the number. Just
as 104 is 1 followed by four zeros, so 3.13 × 104 will be 3 followed by
four other digits (of which the first two are 13).
When the power of 10 is negative, as happens in 2.83 × 10-4, we have
to shift the digits right instead of left: 2.83 → 0.283 → 0.0283 →
0.00283 → 0.000283
In this case, it is actually easier to jump straight to the final answer,
since the negative power (4 in this example) simply counts the zeros to
be stuck on the front, including one zero before the decimal point as
From decimals to standard form
We have seen how to translate standard form into ordinary decimal
notation. Now let’s go in the other direction. Suppose I want to express
2000 in standard form. That’s easy enough: since 2000 consists of two
lots of 1000, or 103, it is equal to 2 × 103. This is now in standard form.
Let’s take another example: 57,800. The definition of standard form
dictates that the answer must look like 5.78 × 10?. The only question is:
what will the power of 10 be? Well, how many times would we need to
shift the digits?
5.78 → 57.8 → 578 → 5780 → 57,800
There are four rightward shifts there, so the answer must be 5.78 × 104.
Alternatively, we could just notice that the original number is a 5
followed by 4 other digits.
The principle is the same for small numbers, such as 0.0000997. Again
the standard form representation will be 9.97 × 10-?. We just need to
know the negative power of 10. It turns out that five rightward shifts are
needed: 9.97 → 0.997 → 0.0997 → 0.00997 → 0.000997 → 0.0000997
Alternatively, we could observe that there are five zeros at the beginning
of the number (including the one before the decimal point). So the
answer is 9.97 × 10-5.
Using standard form for measurements
Using standard form, we can measure any distances in meters. For
instance, the distance to the Sun’s nearest neighbor, Proxima Centauri, is
around 4 × 1016 meters. We could write this as 40 petameters, but this
might raise a few eyebrows, although it is perfectly correct. (It would be
more usual to say 4.2 light years, with one light year coming in at just
under 10 petameters.) But actually, “4 × 1016 meters” is already a
perfectly good description.
Similarly we can measure geological timescales in seconds, if we like:
the Jurassic era began around 6.3 × 1015 seconds ago.
Sum up Powers of 10 are extremely convenient for writing down
very large or very small numbers!
1 Write these numbers out in full.
a 7 million
b 8 billion
c 9 trillion
d 10 quadrillion
e 11 quintillion
2 Express these quantities using suitable prefixes.
a A distance of 18,000 meters
b A computer screen containing 37,000,000 pixels c A blood cell
which weighs 0.000000003 grams d A steam-hammer which
exerts a force of 900,000 newtons e A music player whose
memory is 8,000,000,000 bytes 3 Convert these numbers to
words (such as a millionth) and also decimals (such as
a 10-3
b 10-2
c 10-5
d 10-7
e 10-12
4 Convert these standard form numbers to ordinary decimal
a 6 × 105
b 2.1 × 104
c 8.79 × 10-6
d 1.332 × 10-3
e 6.71 × 1010
5 Write these numbers in standard form.
a 800,000
b 56,000
c 0.00062
d 987,000,000
e 0.00000000111
Roots and logs
• Understanding what roots are and what they are useful for
• Getting to grips with what logarithms actually mean
• Knowing how to switch between the language of powers, roots and logs
A number, when squared, produces 9. What is that number? We could write
this question as?2 = 9. So long as we remember what squaring means (see
Powers if you don’t!), the answer should be obvious: 3 squared is 9 (32 = 3
× 3 = 9), so the answer is 3. In this chapter, we will be interested in this
process of squaring backward. The technical term for this is square-rooting.
So we say that 3 is the “square root” of 9.
Square roots
There are various ways to write a square root: the commonest is to use
the symbol
. (Don’t get this confused with the symbols sometimes
used to write out long division.) So we would write
The easiest way to figure out a square root is to flip the question round,
to talk about squaring instead. So if we’re asked to calculate
, the
answer is going to be the number which when squared produces 49. So
we need to solve?2 = 49.
This question is easy enough (I hope!). But in most cases, the square root
of a whole number will not itself be a whole number. In such cases, you
will need to use a calculator to get at the answer, and there is a
dedicated button
to type
to do the job. So, to calculate
, you would need
, to arrive at an answer of 3.87 (to two decimal
Roots, roots, roots!
It is not only squaring which has a corresponding root. We can ask
exactly the same thing for other powers. For example, if a box is cubeshaped and has a capacity of 64cm3, how wide is it? Well, the volume of
the box must be the width cubed (that is, multiplied by itself three
times). So what we want is the number which when cubed gives 64, that
is, ?3 = 64. This amounts to finding the cube root of 64. The answer is 4
because 43 = 4 × 4 × 4 = 64. We write this as
, introducing a
little 3 to the root symbol.
Then the same thing works for all higher powers too. We could ask for
the fourth root of 81, or
, meaning the number which when it is
multiplied by itself four times gives 81.
According to this rule, where a cube root is written as
root as
, the square root symbol
, and a fourth
could equally well be written as
. But, because it is the commonest root, it is usual practice to leave
out the little “2.”
Most roots of most numbers do not produce a whole number as the
answer. So, often, the safest recourse is to use the calculator. But
beware: the square root button
does not do the job for higher roots!
There is another button for calculating higher roots, which might be
indicated by
. You might also need to press the
key to access this.
So, to calculate
, for, example, you would need to press
to arrive at an answer of 1.55, to two decimal places.
Fractional powers—another way to write roots
There is another way to write roots, which does not use the root symbol
, and which is worth being aware of. We can write roots as fractional
powers. Instead of writing
would write
we would write 4 , and instead of
, we
. Each time, the little number in the root symbol is written
underneath a 1 (as a fraction) and then becomes a power.
The advantage to this is that it allows roots and powers to be combined
quite easily. For instance, you might want first to take the cube root of
8, and then square the result. This looks cumbersome using the root
. In the power notation this can be written much more
neatly as
. (This is based on the second law of powers: (ab)c = ab × c:
see Powers.)
When faced with something like
. there are two things happening to
the number 16. The little 2, at the bottom of the fraction, indicates not
squaring but square rooting. Meanwhile the little 3 (on top of the
fraction) indicates cubing (raising to the power of 3). To calculate the
answer, we perform both of these steps. First calculate the root
Next raise that number to the power of 3, 43 = 64. (In fact, the order
doesn’t matter, you could equally well calculate the power first: 163 =
4096, and then take the root,
. The fact that the two answers
match, and mesh so well with the second law of powers, is what makes
this notation very satisfying!)
Lovely logarithms
There are some words in mathematics which strike fear into the soul,
conjuring up the image of something unimaginably technical and
incomprehensible. One such culprit is the word “logarithm.” But, in
truth, logarithms (or “logs” to their friends) are much tamer creatures
than their fearsome reputation suggests. They are just the opposites of
powers, in the same way that subtraction is the opposite of addition, and
division is the opposite of multiplication.
But how can this be true? Aren’t roots the opposite of powers, as we
have just seen? Yes they are! And yet roots and logs are not the same
thing. In fact, the best picture is to see powers, roots and logarithms as
three corners of a triangle.
To answer the question 2? = 8 is to reason about logarithms. In this case
the answer is 3, and we say that 3 is “the logarithm of 8 to base 2.” This
is written as “log28 = 3.” Although this looks complicated, the meaning
of this expression is exactly the same as “23 = 8.”
The key to answering questions about logarithms is to translate them
into the more familiar “powers” notation. So, when faced with a
challenge such as to find log39, the first step is to translate it into a more
comfortable form: 3? = 9. So the question “find log39” just means “how
many times do we need to multiply 3 by itself to get 9.” When
translating between logarithms and powers, the base of the logarithm
(that’s 3 in this example) is the number which gets raised to a power.
The challenge is to figure out what the power is.
As you might expect, many questions about logs need a calculator. But
beware! Some calculators have several buttons related to logs, and some
have none. A particular warning is that the button simply marked
usually means “log to base 10,” that is, log10.
The general log button is likely to be marked
. However,
not all calculators have this button; on some calculators you cannot
calculate general logarithms directly, only those to base 10.
So powers, roots and logarithms are all different perspectives on the
same idea, namely expressions involving three numbers, such as 23 = 8.
Whether we want to use a power, a root or a log depends on which two
numbers are given, and which is left to be calculated.
• You might be asked “23 = ?.” This is a straightforward question
about powers.
• If you need to answer “?3 = 8” this is a question about roots, and
has the same meaning as
• If you are faced with “2? = 8,” that is to say, “log28 = ?,” this is a
question about logarithms.
We can put these three possibilities in a table. Whether we want a
power, a root or a log is a question of which two of the numbers are
given, and which is left to be calculated.
The usefulness of logarithms
There is one fact about logarithms which made them very useful before
the invention of the calculator: when you multiply two numbers
together, and then take the logarithm, this is the same as adding the two
logarithms of the original numbers together.
To put it another way:
log(x × y) = log x + log y (The law of logarithms)
This is true for any two numbers x and y. I have left the base off the
logarithms here because it doesn’t matter what it is, so long as the three
logarithms have the same base.
This means that when we are faced with something like log311 + log32,
rather than working out the two logarithms separately, we can
immediately combine them into a single calculation: log322
This prompts two questions: why should this be true? And who cares?
The reason it is true follows from the first law of powers which we met
in an earlier chapter. This says that for any three numbers ab × ac = ab
+ c.
If we take logarithms to base a, we get the law of logarithms.
(For the more ambitious reader, here is the argument: If x = ab and y =
ac, then, taking logarithms to base a, it follows that b = log x and c =
log y. Also, we know from the first law of powers that x × y = ab+ c.
Taking logarithms of this, we get log (x × y) = b + c, which says that
log (x × y) = log x + log y.
The mathematics of years gone by
The law of logarithms has been remarkably important in the history of
science and technology. The reason is that it converts questions of
multiplication (which are potentially very tricky), into questions of
addition (which are much easier). Before pocket calculators, the
standard piece of mathematical equipment was a book of log tables,
which listed the logarithms of lots of numbers, to a fixed base (such as
To multiply two large numbers such as 187 and 2012, the procedure was
as follows: look up the logarithm of each number in the book of log
tables. (Any base will do, so long as we are consistent with our choice.
Let’s take base 10.) These numbers have logarithms 2.27184 and
3.30363, respectively (to five decimal places). Next we add these to get
5.57547. To find the final answer, “undo” the logarithm (that is
105.57547) by looking it up in the log tables to find the number that has
this as a logarithm. This gives a final answer of 376,244. You can check
that this is correct!
Sum up Powers, roots and logarithms are close cousins. If you
understand one, you understand them all, so long as you can
remember how they are related!
1 Find these square roots.
2 Use a calculator to work these out to two decimal places.
3 Work out these fractional powers.
4 Find these logarithms.
a log636
b log381
c log464
d log2128
e log5125
5 Simplify these logarithms (to get a single answer such as log1112).
a log46 + log48
b log313 + log32
c log109 + log108
d log57 + log56
e log611 + log610
Percentages and proportions
• Understanding proportions
• Translating between percentages, decimals, fractions and ratios
• Calculating percentages of quantities
Every time you open a newspaper, or watch the news on TV, there is a
certain type of number that you are guaranteed to see. This is the
percentage, indicated by its own little symbol “%.” What is a
“percent”? It is a hundredth, nothing more, nothing less. To say that
you have eaten 75% of a sandwich is exactly the same thing as saying
that you have eaten 75 hundredths of it.
Percentages act as a convenient scale for measuring how much of
something you have, with 0% meaning none of it, and 100% meaning all
of it. Halfway along is the 50% mark.
We have already met another way of writing hundredths, namely
decimals (see Decimals). The percentage 13% stands for thirteen
hundredths, or
and the decimal 0.13 means exactly the same thing.
As this example suggests, translating between percentages and decimals
is straightforward. It is just a question of multiplying (or dividing) by
100, which simply means shifting the digits by two steps, relative to the
decimal point. (Have a look at the chapter on decimals for a reminder of
how to this works.)
So 99% is the same as 0.99, while 76% is the same thing as 0.76, and
8% is the same thing as 0.08. The last of these is the only one where a
mistake might be made; you might think that 8% is the same thing as
0.8, rather than 0.08. But remember that the first column after the
decimal point is the tenths column, and the second one is the hundredths
column. As 8% means exactly “8 hundredths,” it must be equal to 0.08.
On the other hand 0.8 represents the same thing as 80%.
The same system works even when the percentage itself has some
decimal places, as in 64.3%. To change this to a decimal is simply a
matter of moving the figures two steps to the right, relative to the
decimal point: 0.643.
Calculating percentages: from percentages to decimals
How much is 75% of a sandwich? Suppose your sandwich originally
weighs 100 grams. Then it is easy to work out how much 75% is: 1% is
the same as one hundredth of the sandwich, that is, 1 gram. So 75% is
just 75 grams. If the sandwich was 200 grams, then 1% would be 2
grams, and 75% would be 150 grams.
Usually, of course, the figures are less neat than this. What if we want to
work out 13% of a 267 gram sandwich?
To calculate 13% of 267, the first step is to translate 13% into a decimal:
0.13. The second step is then to multiply 267 by 0.13 (remembering that
“of” means multiply). If we have a calculator, this is easy enough
(otherwise we can use a method for doing it by hand; see Decimals).
Some calculators come with a special
button. How does this work?
Not very well, in my experience! I would advise against using it.
Proportion: from fractions to percentages
Percentages are just one way of measuring proportion, that is, the relative
sizes of two quantities. We have already met two other ways: decimals
and fractions. If there are 25 people in a room, of whom 13 are female,
what percentage does this represent? It is easy to represent this as a
of the people in the room are female. To convert this to a
decimal, multiply top and bottom by 4 to get an equivalent fraction
expressed in hundredths (see Fractions). This gives
. The final
step, then, is to convert this to a percentage by moving the digits two
places with respect to the decimal point: 52%.
It is important to be able to translate between the three languages of
percentages, fractions and decimals.
Percentage increase and decrease
Some of the most misunderstood statistics have to do with percentage
increase and decrease.
Suppose that some new houses get built on my street, and 50 new people
move in, raising the number of residents from 200 to 250. What is the
“percentage increase”? Well, the increase clearly amounts to 50 people.
So, to answer the question, we need to know what 50 is as a percentage
of 200. Notice that it is 200, not 250, here: the increase is always gauged
against the starting value. Well, 50 is one quarter of 200, which is 25%.
So the population of the street increased by 25%.
The rule here is:
Percentage increase or decrease is the difference between the old and new
numbers, as a percentage of the old number.
There is a warning which accompanies all these types of figures: they
can feed into a brand of sensationalism beloved of newspaper headline
Suppose your newspaper declares that “Cases of Hill’s disease up 400%
this month!.” This sounds like a catastrophe in the making, conjuring up
images of an epidemic exploding out of control. But the actual figures
might be rather more prosaic. It might be that one person caught the
disease last month, and five caught it this month. That corresponds to a
400% increase, and yet might be within the usual range of variation. If
the average (that is to say mean, see Statistics) monthly figures are three
cases per month, then last month was a little low, and this month is a
little high without being a major cause for concern. Of course, taken
across the population as whole (around 60 million in the UK), the risk
remains vanishingly small. The lesson here is:
Warning! Percentage increases and decreases provide no information
whatsoever about how widespread a phenomenon is!
They are not even particularly useful in describing how the figures are
changing, since an increase of 1000 cases might represent an increase of
200% (if the numbers have risen from 500 to 1500), or equally an
increase of 0.1% (if the numbers have risen from 1,000,000 to
1,001,000). For these reasons, statisticians often complain about the
overuse of “percentage increase,” and similar statistics. It would be more
informative to report the actual figures, rather than sensational statistics.
Percentages provide one way to express proportions. Decimals and
fractions are other possibilities. Now we will have a look at yet another.
Suppose I am mixing a cocktail, and a recipe tells me to mix orange juice
and vodka in the ratio of 3:1. How do I know how much of each
ingredient to pour? In particular, suppose I want the final drink to fill a
200ml glass. How much orange juice and vodka do I need?
A recipe might express this as combining “three parts orange juice with
one part vodka.” This way of expressing the recipe gives a clue how to
calculate the quantities. The crucial observation is that the mixed drink
will consist of four parts in total.
Once we realize this, finishing off is fairly easy. The four parts in total
should amount to 200ml. So one part must be 200ml ÷ 4 = 50ml. We
want three parts of orange juice, which means 3 × 50ml = 150ml, and
one part of vodka, meaning exactly 50ml.
Why did the recipe use the language of ratios instead of just saying
150ml juice and 50ml vodka? The answer is that ratios can easily be
scaled up. If instead we wanted to mix a 2 liter jug of cocktail, the same
ratio 3:1 remains valid, even though the exact quantities change. This
time one part becomes 0.5 liter, so the final mixture is 1.5 liters of juice,
and 0.5 liter of vodka.
Let’s apply the same line of reasoning to a more complicated ratio.
Suppose another cocktail requires tomato juice, lemon juice and vodka
in the ratio of 5:1:2. This time there are going to be eight parts in total,
because 5 + 1 + 2 = 8. We could express the recipe as fractions: it
should be
tomato juice,
lemon juice, and
To convert a ratio into fractions, the rule is:
Add up the total number of parts in the ratio. This goes on the bottom of all
the fractions, and the original numbers go on the top.
If we then want to convert ratios into exact measurements, we multiply
these fractions by the total quantity required. So if we want 2 liters of
the tomato-lemon-vodka cocktail, we need to multiply 2 by each of the
fractions ,
and , in turn, to get measurements of 1.25 liters tomato
juice, 0.25 liters lemon juice, and 0.5 liter vodka.
Interest rates
One place that percentages commonly occur is as interest rates. Suppose
you put $100 in a bank account that has an annual interest of 5%. Since
5% of $100 is $5, after one year, your account should have $105 in it.
What about after another year? A common mistake is to believe that,
each year, the amount will grow by $5, producing $110 after two years.
But the amount grows by 5%, not by $5. In the second year, it will grow
by 5% of what was there at the beginning of the year, that is, $105. To
work out 5% of $105 calculate 105 × 0.05 = 5.25. So, after two years,
the account will contain $110.25. What about after 10 years?
To find the total at the end of a year, we first multiplied the amount of
money at the start of the year by 0.05 and then added the result to the
starting figure. A shortcut is to multiply the figure at the start of the year
by 1.05. So, after one year, the account contains 100 × 1.05. After two
years, it contains 100 × 1.05 × 1.05, that is to say 100 × 1.052. After
three years, it contains 100 × 1.053, and so on. (Look at Powers if you
need to remind yourself how these work.) So, to answer the question
above, after 10 years, the account contains 100 × 1.0510, which comes
out at around $162.89. (If the amount grew by $5 per year, after 10
years the account would contain only $150, so the difference is
In general, to calculate how much money an account contains, we need
three pieces of data. First we need the original deposit made into the
account. Call this M. Then we need to know the interest rate of the
account, making sure this is expressed as a decimal larger than 1 rather
than a percentage. (An interest rate of 5% or 0.05 would be expressed as
1.05; this means the original amount plus interest.) Call this number R.
Finally we need to know how many years ago the deposit was made.
Call this Y. The formula for the amount of money currently in the
account is:
M × RY
Of course, if tax is deducted, the bank changes its rate, or money is
moved in or out, then the story becomes more complicated!
Nevertheless, this formula captures the fundamental rule of interest
Sum up A decimal and a percentage are almost the same thing;
you just need to multiply the decimal by 100. In the same way,
ratios and fractions are almost the same thing. Beware:
percentage increase or decrease does not tell us anything about
the frequency of a phenomenon!
1 Work out each of these percentages as a number.
a Of a pack of 15 dogs, 20% are spaniels.
b In a factory of 400 workers, 77% work full-time.
c In a town there are 1225 roads, of which 32% are designated noparking.
d In a colony of 134,550 ants, 88% are workers.
e A human brain contains 160 billion cells, of which 54% are
2 Express each of these proportions as a percentage, to the nearest
a A packet contained 22 sweets of which 7 are left. What
proportion are left?
b An office has 86 workers, of whom 14 are off sick. What
proportion are off sick?
c 197 of the 458 houses in a village are thatched. What proportion
are thatched?
d In a city of 732,577 people, 118,504 are children. What
proportion are children?
e In a country of 8 million people, 0.5 million wear glasses. What
proportion wear glasses?
3 Convert these percentages to both decimals and fractions.
a 20%
b 100%
c 99%
d 5%
e 4%
4 Find the percentage increase or decrease.
a The number of people over 100 years old was 40 last year and
42 this year.
b The number of cars in my road was 31 last year and 38 this year.
c A girl is 0.74 meters tall one January, and 0.98 meters the same
time next year.
d The number of DVDs sold is 13,488 in December and 11,071 in
e Cases of measles rose from 419 to 1012.
5 Work out the quantities for these recipes.
a Bread dough weighing 1kg is made from flour and water in the
ratio 5:1.
b An exercise regime tells you to go 3 miles, jogging and walking
in the ratio of 4:2.
c Cake mixture weighing 500 grams is made from eggs, butter and
flour in the ratio 1:3:6.
d A drink of 360ml is mixed from fruit cordial and water in the
ratio 1:8.
e A Spanish omelet weighing 440g is made from eggs, onions and
potatoes in the ratio 5:2:4.
6 How much money do you have in each scenario?
a You put $200 in an account at 3% interest, for 5 years.
b You put $50 in an account at 4% interest, for 15 years.
c You put $3 in an account at 20% interest, for 10 years.
d You put $1000 in an account at 3.5% interest, for 30 years.
• Realizing what it means when letters appear in equations
• Understanding how algebra can represent real-life situations
• Learning the rules for working with algebra
For many people, the moment when mathematics moves from being
fairly simple to being incomprehensible is when letters start appearing
where previously there were only numbers. This is algebra. In this
chapter we will have a look at it. We’ll see what it means, and how to
do it without getting confused. Most importantly, we will see why it is
Let’s start with an example. Suppose a restaurant bill comes to $40, and
is to be divided between 8 people. What calculation do we have to do to
work out how much each diner pays? The answer is
were only 6 people? Then the answer is
actually $140? Then the calculation is
. But what if there
. And what if the bill was
. Each of these produces
different answers: as the numbers we put in change, so do the numbers
we get out.
Yet there is a sense in which they are actually all the same calculation.
Each time, the total bill is divided by the number of diners. We could
write this as:
This has an advantage over the previous versions as it makes explicit
what is going on, what principle is being applied here. So if the numbers
are altered, because of a miscount, or an item being missed from the bill,
the same idea continues to work.
Here’s another example. Suppose I am cooking dinner for a group of
people. How many baked potatoes do I need? I reason that each adult
diner will eat 2, and each child will eat 1, and that I should have 5 as
spares in case anyone wants a second helping. This rule comes out as:
Potatoes = 2 × Adults + Children + 5. Then, when the numbers of
guests have been clarified, I can put this principle into action. Once I
know that there will be 5 adults and 3 children, I can plug these
numbers into my rule to arrive at 2 × 5 + 3 + 5 = 18 potatoes.
The fun of formulae
The discussion so far gives us the idea of algebraic formulae. Even if you
wouldn’t usually write these sort of rules down as I have done above, I
hope you agree that this type of thinking is quite normal and natural.
Well, this is algebra. The only difference when experts do it is that,
instead of writing words in their mathematical expressions, they usually
cut down to single letters. What is the point of this? To make things neat
and tidy, and to save space, of course! (It has not always been thus:
mathematicians of bygone eras often wrote lengthy prose in amongst
their equations.)
So, in the potato calculation above, I might begin by calling the number
of adults a and the number of children c. Then the number of potatoes I
need to cook (call it p) must satisfy:
It is usual to omit the × signs when writing algebra using letters, so we
would write this as:
p = 2a + c + 5
What we have arrived at is a typical example of an equation, or a
formula. The power of this method is that it expresses lots of different
facts in just one line.
Many mathematical facts are expressed in this sort of way. For example,
the area of a rectangle is expressed by multiplying its length by its
width. We might write this rule as A = l × w (where A, l and w stand
for the area, length and width, respectively).
The ability to translate between algebraic formulae and English
sentences is one of the central planks of mathematical thinking, and well
worth spending some time on.
From numbers to letters and back: substitution
We have seen how to turn English sentences into mathematical
formulae. What can we do with these formulae? When all is said and
done, we are probably hoping for a number at the end of the calculation,
rather than a collection of letters and algebraic symbols.
To extract a number from a formula, we first need to know how to feed
numbers into it. If we have the formula p = 2a + c + 5, and we are
further told that a = 5 and c = 3, then we can replace the symbols a
and c with these new values, and then work out the value of p:
p = 2 × 5 + 3 + 5 = 18
What we have done here is to substitute numerical values for some of the
letters, and then work out the final answer.
We also saw above that a rectangle’s area is given by the formula A = l
× w. If a particular rectangle has values of l = 8cm and w = 3cm, then
we can substitute these values into the formula to get an area of A =
8cm × 3cm = 24cm2.
The ability to substitute values into formulae becomes more and more
important in all branches of the subject, as the mathematics becomes
more complex. You might object to the previous examples by saying
“multiply the length by width” is quick and simple enough, and doesn’t
really need to be abbreviated as a formula. But if we want to calculate
the volume of a cone (see Area and volume), the formula “
” is a
lot more concise (and, with practice, easier to read) than writing “to find
the volume, multiply the radius of the base circle by itself, and then by
the length of the cone, then divide by 3, and multiply by the ratio of a
circle’s circumference to its diameter.”
Tidying up algebra
There are various rules that we can use to make formulae simpler.
(These will be invaluable when we come to solve equations later.)
The idea is very familiar, when expressed in terms of numbers: just as
we can add up 2 + 3 = 5, similarly we can add 2x + 3x = 5x and 2a
+ 3a = 5a when letters are involved.
Why should this be so? Think of a number and double it. Then add on
your original number tripled. The answer is five times your original
number. Magic! Hardly. This will always work, irrespective of what
number you choose, and this is the rule expressed by 2x + 3x = 5x. The
x, as we have seen, is standing for any number.
This rule is useful for tidying up, or simplifying, algebra. If we have an
expression such as:
2 + 3x + 5x + 2 + 2x
then it can be simplified by collecting together the plain numbers: 2 + 2
= 4 and collecting together the xs: 3x + 5x + 2x = 10x, to leave us
with a much tidier expression: 4 + 10x.
The same thing works when there are more letters involved. If we are
presented with a + 4 + 2b − 5 + b + 3a, then we can gather the plain
numbers together: 4 − 5 = −1, and the as: a + 3a = 4a and the bs: 2b
+ b = 3b, giving a result of 4a + 3b − 1.
Warning! Simplifying algebra is always a good idea, where possible. But
one of the commonest mistakes is to try to simplify things where it
cannot be done. For example, while b + 2b can be simplified to 3b, if we
are faced with the expression b + b2, there is no way to simplify this. It
is not equal to 2b or 2b2. (Why not? Well, if b = 10, then b + b2 = 110,
while 2b = 20, and 2b2 = 200.) Similarly if we have a + b + ab, this
cannot be simplified, and should be left as it is.
Algebra and brackets
Here’s a trick: Think of a number, any number! Now add 4, and then
double what you get. Now add 2. Next, halve the result, and then
subtract the number you first thought of. And the answer is … 5.
How does this work? It is a simple consequence of the algebra of
brackets, which is what we are going to look at in the final section if this
chapter. We’ll see a detailed explanation later on!
Brackets are useful for avoiding ambiguity when writing out calculations
(see The language of mathematics). But they are even more important
when algebra is involved.
The key insight is this. Suppose I add 3 to 5 and then double the answer.
We might write this as 2 × (3 + 5). It is no coincidence that this comes
out the same as doubling 3 and 5 individually, and then adding together
the two results: 2 × 3 + 2 × 5.
In fact, this is exactly the principle used for doing long multiplication:
that 10 × (50 + 2) is the same as 10 × 50 + 10 × 2. We call this
expanding brackets. The idea is as follows: when you have something
being added (or subtracted) inside a pair of brackets, and something
outside the brackets multiplying (or dividing) the brackets, this is the
same as performing the multiplications (or divisions) individually, and
then adding up the answers.
In algebra, we might write a × (b + c) = a × b + a × c. Using the
convention of omitting multiplication signs, this becomes a(b + c) = ab
+ ac. The great thing is that this is true whatever a, b and c are.
So, if we are faced with 2(x + 3 y), we expand the brackets to get 2x +
6 y. Similarly x(x − 3y) = x2 − 3xy. These are both just special cases of
the general rule.
Let’s go back to the trick we started the section with, and let’s call the
mystery number x. The first instruction is to add 4 to it, giving x + 4.
Doubling that produces 2(x + 4). At this stage, let’s expand this
brackets: 2x + 8. Adding on another 2 gives us 2x + 10. Next we were
told to halve the result, which we can write as
, and again, let’s
expand the brackets, producing x + 5 The final instruction was to
subtract the number we first thought of, which of course is x. But now it
is as clear as day that subtracting x from x + 5 will always leave us with
5. It’s not so much Alakazam as Algebra!
Why not try coming up with some of your own tricks along these lines?
Sum up Algebra is a great language for expressing general rules
and laws. Just remember how to translate between algebra and
1 Turn these statements into algebra.
a The number of animals on the ark is twice the number of species
on Earth. (Let a be the number of animals on the ark and s be the
number of species on Earth.)
b The amount of cake on my plate (c) is two divided by the
number of people present (p).
c The number of hours to cook the meat (h) is one quarter of its
weight in pounds (w) plus an extra half-hour.
d The number of patients in the hospital (p) is four times the
number of doctors (d) plus the number of wards (w).
e The temperature in Fahrenheit (F) is the temperature in Celsius
(C), multiplied by nine, divided by five, and then with thirty-two
2 Substitute the values into the formulae.
a If B = t − s, then what is B when t = 13 and s = 5?
b If
, then what is x when y = 108?
c If
, then what is a when b = 4?
d If
, what is D when y = 10 and z = 16
e If z = x2y, what is z when x = 6 and y = 2?
3 Simplify these.
a a + 3a
b b + 5 + 2b − 4
c x + 4y + 2x − 2y
d 5x + 5 + x − 3a − 5
e x + 3z + 2y + 2z + 2
4 Expand these brackets.
a 4(x + z)
b 2(x + 4)
c x(x − 1)
d x(x − 2y)
e 2x(x − 2y)
• Knowing what equations are
• Understanding what it means to solve an equation
• Getting to grips with techniques for solving equations
As we have seen, algebra is useful for expressing general rules. Its
power lies in the fact that it remains true even when we do not know
the values of all the numbers involved. In formulae the numbers whose
value we don’t know are called unknowns. In some happy
circumstances, a formula might contain enough information to pin
down one of these unknown numbers. Then we can solve an equation.
Finding unknowns and solving equations
A good example of finding unknowns is a think-of-a-number game.
Suppose I think of a number, and double it. If the answer is 6, then what
is the number I first thought of? The answer, of course, is 3. But why?
Because the unknown number was doubled to get 6. So, to find it, we
work backward and divide 6 by 2. By doing the original calculation
backward, we can find out where it started. This whole chapter is about
how to turn calculations around, and undo them.
Algebra enters the fray
What we want is a reliable written method for tackling think-of-anumber questions, and their more sophisticated cousins. We need to
think algebraically. So let’s return to the simple problem above, and call
the original (unknown) number x. The idea is to find out its value. What
we are told is that doubling x gives 6, that is, 2x = 6.
Now, we are aiming for an equation of the form
, because that
straightforwardly tells us the value of x. Currently, the left-hand side of
our equation is not x on its own, but 2x. But if we halve that, we will get
x alone, as we want. So that is the tactic.
Here is the key point: to keep the formula true, if we are halving the left-
hand side, then we must also halve the right-hand side. So we halve 6 to
get 3, and arrive at x = 3. We have solved the equation.
The art of equation solving
Of course, with such a simple example, this all seems very easy. But it
illustrates some basic rules. Every equation asserts something. It says
that its left-hand side is equal to its right-hand side. The nexus of the
equation is the equals sign, which separates the left from the right.
The golden rule for this chapter is to keep the equation balanced. What
that means is that if we divide one side by 4, or square it, or add on 32,
then we must always do the same thing to the other side too, to
guarantee that the equation remains true.
The same principles apply, even when the algebra becomes more
complex. Suppose I think of a number, multiply it by 4, and then
subtract 3. The answer I end up with is 17. What was my original
Have you worked it out? How did you do it? Try to reconstruct the
mental steps which led you to the answer. The point is that the algebraic
trickery we are using here is not magic. It’s just a formalization of the
sort of reasoning that we can all apply quite naturally.
Let’s look at the last puzzle in more detail. The starting point is to
translate it into algebra. We started with an unknown number: call it x.
Then we multiplied it by 4 to get 4x. Next, we subtracted 3 to give us 4x
− 3. Finally, we are told that this number is 17, which produces the
4x − 3 = 17
As before, the aim is to turn this into a very simple equation of the form
, as this directly tells us the answer. With that in mind, the key is
to undo the various steps, starting with the last one performed and
working backward, until we end up with x on its own.
So, the first thing to do is to get rid of the −3. This means adding on 3
and, as always, we have to do the same thing to both sides. Adding 3 to
the left-hand side gives just 4x, which is an improvement on 4x − 3, as
it is simpler. Adding 3 to the right-hand side produces 20. So now we
4x = 20
All that remains is to divide both sides by 4. Dividing 4x by 4 leaves x
alone on the left-hand side, which is exactly what we were aiming for.
Dividing 20 by 4 gives 5. So we end up with:
One of the good things about equations is that you do not need anyone
to tell you whether you have got it right. It is easy to check, and I would
suggest that you always do this. If our solution above is correct, then if
we take the original equation 4x − 3 = 17, and substitute in the value X
= 5, we should get a true statement. If we don’t, then we know we have
made a mistake. So let’s try: 4 × 5 − 3 = 17; this is true, so we have
solved the equation correctly.
If we were presented directly with the equation 4x − 3 = 17, how
would we know where to start: which move needs to be undone first?
The two options are multiplication by 4 and subtraction by 3. An old
friend comes to our assistance: BEDMAS (see The language of
mathematics). That tells us that the multiplication was done first,
followed by the subtraction. So, to undo these steps, we go in reverse
order and tackle the −3 first.
The xs stick together: collecting like terms
Here is another think-of-a-number problem: I think of a number, treble it
and subtract 2. The answer I get is the same as if I double my original
number and add 4.
As usual, let’s call the mystery number x. If we treble it, we get 3x, then
subtracting 2 gives 3x − 2. Now the problem tells us that this is the
same as another quantity, which we get if we double x to get 2x, and
add 4 to get 2x + 4. So the equation which encapsulates this problem is:
3x − 2 = 2x + 4
Now, as always, we are aiming for a final equation of the form
But, this time, things look trickier, as there are xs and numbers on both
sides of the equals sign. The first step, then, is to improve this situation.
We want to eliminate the xs on one side or the other.
The tactic is to collect together the xs on one side, and the numbers on
the other. But how can we do this? Let’s deal with the xs first, and let’s
decide to collect them on the left (we could equally well choose the
right). That means getting rid of all the xs on the right. Well, there are
two of them, that is to say 2x. We can eliminate these by subtracting 2x
from the right. Of course, to keep the equation balanced, we also have to
subtract 2x from the left:
3x − 2 − 2x = 2x + 4 − 2x
(It is not strictly necessary to write this step out; I am just doing it to
make explicit what is happening.) Now on the right-hand side, the 2x
and the −2x cancel each other out as planned, leaving only 4. On the
left-hand side, we start with 3x and take away 2x, leaving just x. So our
equation now reads:
This is much simpler already, and now very quick to finish off. By
adding 2 to both sides of the equation, we get the solution x = 6.
Let’s review that last example: we collected together all the xs on one
side, and all of the plain numbers on the other. This technique is called
collecting like terms, and it is the best way to make a complicated
equation simpler.
There are two other little tricks we might need. Firstly, if A = B then it
is equally true that B = A. This means we can swap over the two sides
of an equation any time we like. So if we find that 12 = 3x, we can
easily swap the sides over and write this as 3x = 12. The second trick
involves flipping +/− signs. We can do this so long as we follow the
golden rule and do it to both sides. So if we have −4x = −8, we can
flip the signs to get 4x = 8. Similarly, if we have −5x = 10, we can flip
the signs on both sides to get 5x = −10.
Some things are more equal than others: inequalities
The equation has a lesser-known cousin: the inequality. Here the nexus is
not the equals sign (=), but one of the four inequality symbols. The first
two are < and >, standing for “is less than” and “is greater than,”
respectively. So we might truthfully write 4 < 7, or write × > 9 to
indicate that x is some number larger than 9. These two symbols
represent strict inequalities. (Alternatively you can think of these two
symbols as being just one reversible symbol: 4 < 7 means the same as 7
> 4.)
The weak inequality symbols ≤ and ≥ stand for “is at most” and “is at
least,” or “is less than or equal to” and “is greater than or equal to,”
respectively. So, while it is true that 4 ≤ 7, it is also true that 4 ≤ 4.
(But it is not true, that 4 < 4.)
With these symbols in place, we can write inequalities in the same way
as we write equations. For example: 5x − 4 < 3x + 2. But what does it
really mean to “solve” an inequality? We cannot hope for a unique
answer. Instead, we want to pin down a range of values of x for which
the original inequality is true.
If we follow the rules above, we first collect the xs on the left, by
subtracting 3x from both sides: 2x − 4 < 2. Next we want to collect the
plain numbers on the right, by adding 4 to both sides: 2x < 6. Finally
we can divide by 2 to get out answer: x < 3. This is the solution to the
inequality, and expresses the full range of values of x for which the
original inequality is true. If x < 3 then it should be true, but if x ≥ 3, it
should not. Test it out!
Changing signs and symbols
Solving inequalities is almost identical to solving equations. But there is a
danger-point where the two diverge. If we find that 8 = 2x, then we can
swap the sides of the equation to get 2x = 8. With an inequality,
however, when we swap sides, we also have to reverse the inequality
symbol. So 8 > 2x becomes 2x < 8.
There is another situation where the inequality has to be reversed. In the
context of an equation, if we find that − x = −7, for example, then we
can flip signs on both sides to positive, to get a final answer of x = 7.
Let’s think about the inequality − x < −7. The value x = 8 does satisfy
this, since −x = −8, which is indeed less than −7. Similarly the value
x = 6 does not satisfy − x < −7, since −6 is not less than −7. If we
simply flip the sign on both sides of the inequality − x < −7 (as we do
with equations), we get x < 7. But we have just seen that this is not the
right answer. The rule is that when you change the signs in an inequality
(or, equivalently, when you multiply or divide both sides by a negative
number) you need to reverse the inequality symbol. So, if we have − x
< −7, we change the signs to positive on both sides, but in doing so we
must also reverse the < symbol, giving an answer of x > 7.
Sum up Solving equations occupies pride of place at the heart of
mathematics, as this is the main tool for getting answers to
algebraic questions. Just remember the golden rule: keep the
equation balanced!
1 Solve these think-of-a-number problems!
a I think of a number, multiply it by 3. The answer is 18. What
was my number?
b I think of a number, multiply it by 4, and then subtract 3. The
answer is 25. What was my number?
c I think of a number, divide by 2, and then multiply it by 3. The
answer is 9. What was my number?
d I think of a number, subtract 4, and then multiply it by 5. The
answer is 20. What was my number?
e I think of a number, subtract 3, and then divide by 5. The answer
is 5. What was my number?
2 Solve the problems in quiz 1 again, using algebra.
3 Solve these equations.
a 4x = 2x + 6
b 3x − 4 = 2x + 8
c 5x − 4 = 14 − x
d 2x + 2 = 12 − 3x
e 1 − x = 8 − 2x
4 Solve these equations.
a 2(x + 1) = 3x
b 3(x − 1) = 2(x + 1)
c 4(x − 2) + 2 = 2(x + 5)
d 4(2 − x) −1 = 2(x + 3) + 1
e 2(1 − x) + x = 3(2 − x) + 2
5 In quiz 3, try replacing the “=” sign with “<” and solving the
resulting inequalites.
• Measuring angles
• Knowing the rules for calculating angles
• Understanding the geometry of parallel lines
This chapter contains truly ancient knowledge. Everything we shall see
here was included in the greatest textbook of all time: the Elements,
written by Euclid of Alexandria around 300 BC. It may seem surprising
that Euclid’s geometry continues to be studied thousands of years later.
But he got to grips with the root ideas of geometry: straight lines and
circles, and the ways they interact. We will explore these in later
chapters, including the many shapes that can be built from straight
lines, of which the most important are triangles. But first we look at the
fundamental geometrical notion of an angle.
An angle is an amount of turn. An ice skater or ballet dancer who spins
on the spot might turn through a large angle. During a long journey, a
car tire is likely to spin through a very large angle indeed! How do we
measure angle? Well, just as distance can be measured in miles,
centimeters or light-years, there are different units for angle too.
The revolution is coming
A common unit of turn is the revolution. If you stand on the spot, make
one complete turn and end up facing in exactly the same direction as
when you started, then you have completed one revolution. In
engineering, the speeds of wheels and cogs are often discussed in terms
of revolutions per minute (rpm). The second hand on a clock has a
rotational speed of precisely 1rpm, while a washing machine drum
might whizz round at 1000rpm.
An even more common unit is the degree. This is generally preferable
when measuring small quantities of turn, less than a single revolution.
One revolution is broken up into 360 degrees (usually written 360°). So
if you turn on the spot until you are facing exactly backward, then you
have turned 180° (that is, half of one revolution).
In many sports, such as skateboarding, ice-skating and diving, turning is
an important ingredient. Typically, the amount of turn is measured in
degrees. So a skateboarder might say that they have performed an “Ollie
180.” This means that they have turned 180°, that is
a revolution.
(What exactly an “Ollie” may be is a matter for another day!) Similarly a
turn of 720° translates to two complete revolutions, because 720 = 2 ×
In terms of the numbers, to translate from revolutions to degrees, we
need to multiply by 360. So
revolution is, in degrees,
Going the other way, to convert degrees to revolutions we need to divide
by 360. So, as we saw, 720° is
It is useful to practice translating between the language of degrees and
revolutions (especially when talking to skateboarders!).
Here is some jargon about angles:
• An angle of 90° (or
revolution) is known as a right angle. Saying
that two lines are perpendicular means that they cross at right
angles. Also notice the little box we draw at the angle, to indicate a
• An angle less than 90° is known as acute.
• An angle between 90° and 180° is called obtuse
• Angles between 180° and 360° go by the name of reflex.
Measuring and drawing angles
There are various pieces of equipment we can use for measuring angles.
One is a compass: not the device for drawing circles, but the type for
finding North. If you want to know the angle a road makes to the North–
South line, a compass will tell you. Compass bearings and angles are
essentially the same thing.
In the context of geometry, though, a more familiar (though lower-tech)
tool is a protractor. It’s a semicircular piece of transparent plastic, with
various markings on it: there is a base-line running along the flat edge
and, at the center of this, a cross-hair; around the curved edge, angles
are marked from 0 to 180°.
How do we use a protractor? If we have two straight lines which meet at
a point, we might want to know the angle between them. What does this
mean? imagine that the two lines are roads, and you are standing at the
point where they meet, looking along one road. What we want to know
is the angle through which you have to turn, so that you are looking
down the other road (line). This is a task for a protractor! Lay it on top
of the page, with its base-line lined up with one of the two lines, and
with the cross-hair sitting exactly on the point where the two lines meet.
Looking at the curved edge of the protractor, you can see the numbered
marks representing angles. The number at the point where the second
line emerges is the number we want. Here is a warning though:
Warning! Protractors usually have two scales: one measuring angles
clockwise, the other counterclockwise. Make sure you pick the one
which counts up from 0°, not the one counting down from 180°!
We can draw angles by adapting this procedure slightly. If we want to
draw an angle of 65°, say, the first thing to do is draw one straight line
in the position that we want. Then, mark the point on that line where we
want the second line to branch off (in this case this is at the end of the
line), and place the cross-hair of the protractor on that point, with its
base-line matching up with the drawn line. Next, find the 65° on the
protractor (bearing in mind the warning above!) and make a mark on
the page at that position. Finally, remove the protractor, and join the
two marked points with a straight line.
Angles—what’s the point?
Look at this picture. It might be showing a road which is perfectly
straight except at one point, where it has a corner. Where the two
straight sections meet, there are two angles. Imagine that you are
standing at the corner, in the middle of the road, facing along one
section of road, and then you twist around, so that you are facing down
the other. There are two ways you could do this: you could rotate
clockwise or counterclockwise, and depending on your choice you will
turn through a bigger or a smaller angle.
Now, these two angles are related in a precise way: if you turn through
one, and then turn through the other, you will have turned one complete
revolution. So, the golden rule tells us that, between them, these two
angles add up to exactly 360°. This means that if we know one of them,
say it is 72°, then we immediately know that the other must be 360° −
72° = 288°.
Angles like this are known as angles at a point, and this line of thought
works just as well when there are more than two involved. We might
think of the picture below as a junction of three roads. But again, the
three angles must add up to 360°. So if two of them are 37° and 193°, we
can immediately calculate the third angle (marked x) as 360° − 37° −
193° = 130°. No matter how many angles there are, the rule is: Angles at
a point add up to 360°.
A similar argument works when we have a situation like the one
opposite. This time we’ve got one straight road, maybe a motorway, and
a turning coming off it. In this picture, though, the two marked angles
do not add up to a full 360°. If you turn through one and then through
the other, you have completed a half turn, which is 180°. Situations like
this are known as angles on a straight line.
Angles on a straight line add up to 180°.
So as soon as we know one of the angles, say it is 35°, then we can
calculate the other: 180° − 35° = 145°.
The attraction of opposites
When two perfectly straight roads cross each other, four angles are
formed. Because these are angles at a point, the four must add up to
360°. But there is something else we can say too: if you look at the
picture below, the angles seem to come in pairs: two large ones and two
small ones.
This is not an illusion, in fact the two small angles must be equal to each
other, as must the two large ones. These are known as opposite angles,
and the rule is:
Opposite angles are always equal.
Why is this true? Well, one large angle together with either of the two
small angles form angles on a straight line, and so must add up to 180°.
That means that the large angle determines both of the small angles,
each by exactly the same calculation. As soon as we know just one of the
four angles, we can quickly work out all the others.
Never meeting—parallel lines
The notion of two lines being parallel is fundamental. It means that they
can be continued forever, in both directions, without ever crossing. To
say the same thing in a different way: the distance between two parallel
lines is always the same, no matter where you measure it. Yet another
way to say this is that the angle between parallel lines is 0°. In
geographers’ terminology, they have the same bearing. Train tracks are a
classic example of parallel lines.
In geometry, lines are indicated to be parallel by drawing matching
arrows on them.
It is often useful to know when two angles are equal to each other. We
have seen some examples above, namely opposite angles. Parallel lines
also present situations where equal angles can be found. Let’s suppose
we have a pair of parallel lines, and a third line which crosses them both
(like a straight stick lying across a pair of railway tracks). In such a
situation, there are two handy rules which tell us which of the resulting
angles are equal.
To start with, angles which are in identical positions, but on the two
different lines, are said to be in corresponding positions. The first rule is:
Angles in corresponding positions are always equal. (Law of corresponding
This should not come as too much of a surprise, since the two angles are
identical in every respect, but are just in different places. Sometimes
they are known as “F” angles, because the pattern looks like a capital F.
(Don’t rely on this, though, as the F may be back to front or upside
down, or otherwise deformed!)
The second law of parallel lines and angles deals with the slightly subtler
notion of alternate angles. Pairs of alternate angles lie on opposing sides
of the third line, and are sometimes known as “Z” angles.
Alternate angles are always equal. (Law of alternate angles)
Why should this be true? It is a consequence of the laws on opposite
angles, and corresponding angles. Try to see why!
Sum up Angles and parallel lines are the salt and pepper of
geometry. Just remember how to combine them, and they will
work wonderfully together!
1 Convert these angles from degrees to revolutions.
a 180°
b 360°
c 90°
d 270°
e 540°
2 Use a protractor to draw two lines that meet at each of these
angles. Next to each angle write whether it is acute, obtuse or
a 45°
b 72°
c 83°
d 112°
e 229°
3 Sketch the following situations and calculate the missing angles.
a Two angles meet at a point. One is 75°. What is the other?
b Three angles meet at a point. One is 110°, another is 40°. What is
the third?
c Two angles lie on a straight line. One is 60°. What is the other?
d Three angles lie on a straight line. One is 111°, another is 32°.
What is the third?
e Four angles meet at a point. Three are 51°, 85° and 190°. What is
the fourth?
4 Sketch the following situations (no need to use a protractor), and
calculate all four angles. In each case, two straight lines cross,
producing four angles.
a One angle is 90°. What are the other three?
b One angle is 45°. What are the other three?
c One angle is 21°. What are the other three?
d One angle is 122°. What are the other three?
e One angle is 176°. What are the other three?
• Recognizing different types of triangle
• Reasoning about a triangle’s angles
• Calculating a triangle’s area
Two straight lines do not make a shape. But three do. This is why the
triangle is one of the most important figures in geometry: it is one of
the simplest.
But is the geometry of triangles useful? The answer is yes, as the world is
full of triangles, even if most of them are invisible. What, you may ask,
are invisible triangles? Whenever you choose three points in space, you
have defined a triangle. For instance, I might pick the place where I am
standing, where my wife is sitting, and the television in the corner of the
room. These three points form a triangle.
In any such situation, the techniques in this chapter can apply, even
when the triangle’s sides are not immediately obvious. For this reason,
triangular geometry really is everywhere, and we shall be meeting more
of it later (see Pythagoras’ theorem and Trigonometry).
Different types of triangle
For many people, one triangle is much the same as any other. But, to the
connoisseur, triangles come in a variety of forms, each with their own
characteristics. The most symmetrical triangle is the one where the three
sides are all the same length. Triangles like this are called equilateral
(coming from the Latin for “equal sides”).
It automatically follows that the three angles inside an equilateral
triangle must also be equal. As we shall see shortly, each of these angles
is equal to 60°. So an equilateral triangle has a very rigid shape, with no
room for maneuver. The only possible variation is in the triangle’s size.
In all other respects, every equilateral triangle looks the same, in much
the same way that all squares look the same. In fact the equilateral
triangle and the square are the first two regular shapes, meaning shapes
whose sides and angles are all equal. We shall explore this further in
Polygons and solids.
A slightly less symmetric type of triangle is one which has two of its
lengths the same. Such a triangle is known as isosceles (coming from the
Greek for “equal legs”). There is more room for maneuver with isosceles
triangles: they can look very different, as we shall see shortly.
Most triangles are neither equilateral nor isosceles, but have three sides
of different lengths. There is a word for this too: scalene (from the Greek
for “unequal”). These words—equilateral, isosceles and scalene—refer to
the lengths of the triangle’s sides. But triangles can also be described by
the sizes of their angles. A right-angled triangle is, unsurprisingly, one
which contains a right-angle. That is to say one of its three angles is
equal to 90°. (These are in some ways the “best” triangles, and we will
be hearing a lot more about them in Pythagoras’ theorem and
Triangles whose angles are all less than 90° are called acute triangles.
(Remember from the previous chapter that an angle less than 90° is
called an acute angle.) Similarly, a triangle which contains an obtuse angle
(that is to say one that is more than 90°) is known as an obtuse triangle.
That’s a large amount of jargon to digest, and all just to describe
different sorts of triangle!
Angles in a triangle
The starting point for the geometry of triangles is a relationship between
the three angles inside the triangle. Try drawing a triangle which
contains two right angles, or two obtuse angles. You will fail. Why? The
answer is given by this chapter’s golden rule.
This rule puts a limit on the possible sizes of the angles inside any
It will prove useful for calculating the sizes of angles in triangles. But
why should it be true? It is worth seeing a proof of this famous fact,
since it is not complicated. Indeed, it follows from the facts about angles
and parallel lines that we saw in the previous chapter.
We begin by taking a triangle, call it ABC, meaning that the three
corners are named A, B and C respectively. (These are just labels so that
we can refer to them individually.) At each corner is an angle. Since we
don’t know their values, we had better name these too. Let’s call them a,
b and c. So what we want to show is that a + b + c = 180°.
We also have the three lines of this triangle, which we might call AB, BC
and CA, where AB is the line running from corner A to corner B, and so
The main move in the proof is this: we add in a new line, parallel to AB,
which passes through the point C.
Now there are now three angles fanning out around C. What is more,
these three angles must add up to 180° as they are angles on a straight
line, in the terminology of the previous chapter.
The middle angle c is unchanged. The insight is that one of the other two
is actually equal to a; this follows from the rule of alternate angles (or “Z
angles”) that we saw in the last chapter. By exactly the same reasoning,
the third angle must be equal to b. Now we have finished! The three
angles at C are a, b and c, but these are angles on a straight line, and so
a + b + c = 180°.
Triangular angular calculations
Now that we know the three angles in a triangle always add up to 180°,
we can use this to perform some calculations.
To start with, if we know two angles of a triangle, we can always work
out the third. For instance, if a triangle contains angles of 30° and 45°,
then the final angle must the number which when added to 30° + 45°
gives 180°. That is to say the final angle must be 180° − 30° − 45° =
In some cases we can do better than this. If the triangle is equilateral,
then we know immediately that all three angles are equal. So it follows
that each must be 180° ÷ 3, that is, 60°.
Similarly, if a triangle is isosceles, we can often work out its angles quite
quickly. In isosceles triangles, two of the three angles are equal. So now
we just need to be given one to work out the others. Suppose ABC is an
isosceles triangle where the lengths AB and AC are the same. As usual,
we’ll call the three angles a, b and c. It follows that the angles at B and C
must also be equal, that is to say, b = c. Suppose we are now told that b
= 55°. Then we know immediately that c = 55°. Finally we can work
out angle a. It is 180° − 55° − 55° = 70°.
Areas of triangles
We are now going to look at the area of triangles, that is, how much
space there is inside them. (We will study area more generally in Area
and volume.)
There is a nice rule for calculating the area of a triangle: multiply the
base of the triangle by its height, and then divide by 2. So the formula is:
Or, if we call the area A, the base b and the height h, the formula
This rule is very convenient, and easy to use. But it comes with a few
words of warning nevertheless!
Firstly what do these terms mean? What is the “base” of a triangle? It is
the side which runs along the bottom. All right, but here’s something to
think about: if we spin the triangle around, it will take up the same
amount of space. So the area won’t change. But the base does change!
The edge which was at the bottom is no longer the base and one of the
other sides takes its place.
So what’s going on? In fact, any of the three sides will do as the “base.”
So this is really three formulae in one, depending on which side you
Let’s suppose we have chosen one side as the base. Now, what is the
“height”? This is where mistakes often get made! The height is the
vertical distance from the base to the top corner of the triangle. That
sounds reasonable, I hope, but beware:
Warning! The triangle’s height may not coincide with any of its edges!
Only in right-angled triangles is the height of the triangle the length of
one of its sides. In all other triangles it isn’t. The rule is that the height
must always be measured at right angles to the baseline: it is the vertical
height from the baseline to the top corner. In fact, the height-line may
not even be inside the triangle, as this picture shows!
Let’s have an example. Suppose I have decided to look for a house to
buy, in the region between three towns: Asham, Bungleside and
Cowentry (A, B, C). The question is: how large is the area I have to
search? Using a map, I measure the distance from A to B as 50 miles.
Taking this as the base, then the height must be the perpendicular
distance from the line AB up to the third corner C. The map says that
this is 60 miles. So the area of my triangle is
square miles.
The triangle within
The formula for the area of a triangle is undeniably useful. But why is it
true? Have a look at the triangle below. There is the baseline (b), and a
height-line (h). Notice that the height-line divides the shape into two
smaller triangles. (As it happens, these are both right-angled triangles.)
Now, we can fit the whole triangle inside a rectangle. What is more this
rectangle has the same basic dimensions as our triangle. Its height is h,
and its width is b. So the area of the rectangle must be b × h. (See Area
and volume for more on this.)
What I want to show is that the triangle takes up exactly half the space
inside the rectangle. Why? Well, look at the two little triangles. There is
an exact copy of each of them inside the rectangle. So the rectangle is
divided into four triangles: two copies of each of the two small triangles,
of which one is inside and one outside the original triangle. So the
original triangle is exactly half of the rectangle, and its area must be
as expected.
Sum up Triangles are the simplest shapes you can build with
straight lines. But triangles can look very different! Luckily,
there are some elegant rules for finding their angles and their
1 Draw a triangle, as accurately as possible and reasonably large,
to fit each of these descriptions.
a An equilateral triangle
b An isosceles right-angled triangle
c A scalene obtuse triangle
d An isosceles acute triangle
e An isosceles obtuse triangle
2 Measure all the angles of each of the triangles you drew in quiz
1. Check that the three angles sum to 180°.
3 Sketch the triangles and calculate the angles.
a If a right-angled triangle also contains an angle of 30°, what is
the third angle?
b ABC is a triangle where the angle at A is 120°, and the angle at B
is 45°. What is the angle at C?
c ABC is an isosceles triangle, where AB = AC. The angle at B is
70°. What are the other two angles?
d ABC is an isosceles triangle, where AB = BC. The angle at B is
70°. What are the other two angles?
4 Sketch these triangles and calculate their areas.
a A triangle with base 2cm and height 1cm
b An acute triangle with base 2cm and height 2cm
c A right-angled triangle with shortest sides of 3cm and 4cm.
d A triangle in which the longest side is 5cm, and the
perpendicular height from that side is 1cm
e A triangle whose base is 6cm and height is 3cm, where the
height-line runs outside the triangle
• Knowing the parts of a circle and how they are related
• Meeting the famous and mysterious number π
• Understanding the meaning of the formula πr2
Geometry is full of beautiful shapes, but it has often been said that the
simplest and most bewitching of them all is the circle. Certainly this is
among the most useful shapes for humans, and has been since the
invention of the wheel many thousands of years ago.
There is much than can be said about the geometry of circles. In fact,
some of the most famous formulae in science are about circles. In this
chapter we will explore these mathematical marvels.
To begin at the beginning: what is a circle? The ancient Greek geometer
Euclid defined it this way: first pick a spot on the ground. That will be
the circle’s center. Now choose a fixed distance, say 5 feet. Then mark
every point on the ground which is exactly 5 feet from the center. The
shape that emerges is a circle.
The language of circles
Circles have their own little lexicon of terms that you need to get to
grips with. To begin with, the radius of a circle is the distance between
the center and the edge of the circle. (That was 5 feet in the example
A compass is a useful tool for drawing circles (also known as a pair of
compasses: that’s the tool with a pin and pencil, not the one for finding
North!). If you have a compass, then the distance you set between the
pin and the pencil will be the radius of the resulting circle.
Another key word is diameter. This is the distance across the circle, from
one side to the other, passing through the middle. A little thought should
confirm that a diameter can be split into two radii, meeting in the
middle. This gives us our first formula for the circle: if d is the diameter
of a circle and r is its radius, then the number d is r doubled. Or more
concisely, d = 2 × r. Omitting the multiplication sign, as usual, we
d = 2r
So a radius of 5cm corresponds to a diameter of 10cm, a diameter of 6
miles corresponds to a radius of 3 miles, and so on.
π: the legend begins
So far, so easy. But there is another significant distance which we might
be interested in: the length of the circle itself, that is to say the
circumference. If I have a circular pond in my garden, with a fountain at
the center, and a radius of 1 meter, how long is the wall around the
outside of the pond?
This question, of finding the circumference of a circle when we know the
radius and diameter, is a remarkably deep one, and has baffled thinkers
around the world since the dawn of civilization. The ancient
Babylonians, almost four thousand years ago, thought that if you
multiplied the diameter by 3.125 (that is,
) you would get the length of
the circumference. Egyptian thinkers around 1650 BC believed that the
value should be
(which is around 3.160). In China around AD 500 Zu
Chongzhi settled on a figure of
, while al-Khwarizmi in ninth-century
Baghdad believed it should be
All these geometers did at least agree on one thing: there is some
number which works for all circles, however large or small. When you
multiply the circle’s diameter by this mystery number, you get the
circumference. The only difficulty was in identifying this number
It was in 1706, at the hands of Welsh mathematician William Jones, that
this elusive number finally received the name by which it is now
universally known: π. Pronounced “pi,” this symbol is the Greek letter
“p” (probably chosen to stand for “periphery”).
In the centuries that followed, we have learned a great deal about π. In
particular, we now understand why the geometers of old struggled with
it so much. The number π is an example of what is today known as an
irrational number. This means that it can never be written exactly as a
fraction of two whole numbers. (“Irrational” here means “not a ratio”; it
has nothing to do with rationality in the sense of being sensible,
intelligent or logical.)
This immediately means that all the old values attributed to π must be
wrong because they were fractions (although some were excellent
estimates, and were near enough for practical purposes). What about a
decimal? We can certainly start writing out the value of π:
3.1415926535897932384626433832795 …
The interesting thing is that this sequence of numbers will continue
forever, never ending, and never getting caught in a repetitive loop
(unlike the recurring decimals we came across when converting fractions
to decimals). It simply keeps going, ever unpredictable. Hence we can
never write down the value exactly, except, of course, under its name:
“π.” (The drive to calculate ever more digits of π has now reached the
trillion mark, and, like π itself, is set to continue indefinitely.)
All the way around: the circumference
Interesting though the history is, the mystery of π is now largely solved.
In particular, it is now easy to calculate the circumference of a circle
from its diameter: you simply multiply by π. So we get the next formula
for a circle: c = π × d, where c is the circumference and d is the
diameter. Equivalently, because the diameter is twice the radius (r), we
might say c = 2 × r × π, or omitting the multiplication signs, and
c = 2πr
For most of us, the correct tool for using this formula is the
button on
a calculator. Though not exact, this will give as good an approximation
of π as we will ever need. (In some calculators you may have to press
SHIFT or 2nd FN and then another button to access the π function.) So,
to return to the example of the pond above, if my circular pond has a
radius of 1 meter, then its circumference is 2 × π × 1. Typing in
to my calculator produces an answer of 6.28 meters (when
rounded to two decimal places; see Decimals).
Turning this round, if we know that a circle has a circumference of 10
meters, how can we calculate its radius? In other words, we have to find
the value of r so that 2 × π × r = 10. To solve this, we just need to
divide 10 by 2 × π (see Equations if you have forgotten why this works).
On my calculator I do this by typing
but other
calculators may work differently. This brings up an answer of 1.59
meters, to two decimal places. (The brackets are needed to make sure
that I divide 10 by twice π, rather than dividing by 2 and then
multiplying the answer by π.)
So now you know how the radius, diameter and circumference of a circle
are all related.
We might also want to know the area of a circle, that is, the amount of
space it occupies. For some shapes, such as a square, finding the area is
easy: if the square’s side is 3cm long, then its area is 3 × 3 = 9cm2.
Once again, though, the circle is less straightforward, and again the
number π takes center stage.
If a circle has radius r = 3cm, then what is its area? If we build a little
square on the radius of the circle, we know that its area is r × r, or r2
for short. That comes out as 3 × 3 = 9cm2 in the above example.
So, the question is, how many times does this little square fit inside the
circle? The answer—for any circle, large or small—is again π. So the
area of the circle is given by π × r × r. Calling the area A, this gives us
one of the most famous of all formulae:
A = πr2
In the example above, where the circle has radius 3cm, the area is π × 9
= 28.27cm2, to two decimal places.
Going backward
We might want to work backward. If we know the area of a circle, how
can we work out its dimensions? Suppose a circle has an area of 4cm2. If
we call its radius r, it must be that πr2 = 4. This is now an equation,
where our job is to find out r. (See Equations for more discussion of this.)
We begin by dividing both sides by π, to get
. We could now put
this into the calculator, but let’s first see how to finish the calculation
off. We now know what r2 is. To find out r from this information, we
need to take the square root (see Roots and logs) of the number we have
just found. So the exact answer is
There are various ways to get a value for this, depending on your
calculator. One way would be
. Again the brackets
are crucial. On an older calculator, the best approach might be
followed by
(or maybe
, where
is the
button which recalls the answer to the previous calculation). Whatever
method you use, the answer should come out as 1.13cm, to two decimal
Sum up Circles are among the most beautiful and useful shapes.
They are also associated with some of the most beautiful and
useful of all mathematics!
1 Use your compass to draw circles with these sizes. Then, for each
circle, estimate the length of the circumference (that is, the
distance around the outside).
a A circle with a radius of 2cm
b A circle with a diameter of 2cm
c A circle with a radius of 4cm
d A circle with a diameter of 4cm
e A circle with a radius of 1cm
2 Calculate these lengths. Give your answers to two decimal places.
a The circumference of a circle with diameter 5cm
b The circumference of a circle with radius 5cm
c The radius of a circle with diameter 7 miles
d The diameter of a circle with circumference 50km
e The radius of a circle with circumference 4.4mm
3 Calculate these areas. Give your answers to two decimal places.
a The area of a circle with radius 5cm
b The area of a circle with diameter 5cm
c The area of a circle with radius 2.13mm
d The area of a circle with diameter 2.13mm
e The area of a circle with circumference 2.13mm
4 Calculate these.
a What is the radius of a circle with area 5 square miles?
b What is the diameter of a circle with area 13 mm2?
c What is the circumference of a circle with area 5.3km2?
d A circle fits inside a square exactly (touching but not crossing all
four sides). If the area of the circle is 8cm2, what is the area of
the square?
e Overnight, a pattern of flattened crops appears in a farmer’s
field, consisting of four non-overlapping circles, all the same size.
If the total area of flattened crops is 700m2, what is the radius of
each circle?
5 Concentric circles are circles with the same center.
a A picture consists of two concentric circles with radii 3cm and
4cm respectively. What is the area of each?
b The circular strip between the two circles in part a is painted
red. What is the area of the red stripe?
c A circle fits inside a square exactly. The width of the square is
5cm. What are the areas of the square and the circle?
d In part c what is the total area of the parts of the square outside
the circle?
e Inside a circle of radius 6cm is a triangle, whose base is a
diameter of the circle, and whose height is a radius. What is the
total area of the parts of the circle outside the triangle?
Area and volume
• Understanding how length, area and volume are related
• Knowing how to calculate areas and volumes for different shapes
• Getting to grips with the units for area and volume
We have many units for measuring distance: miles, millimeters, yards,
parsecs, furlongs, inches, light-years, meters, feet, … and that ignores
archaic measurements such as rods, leagues, paces and perches. In this
book we will mostly stick with meters, and related units such as
millimeters, centimeters, and kilometers (see The power of 10). Any
measure of distance automatically produces a related measurement of
Distance is one-dimensional: it measures a length along a line. Area, on
the other hand, is two-dimensional. It measures the size of a surface.
What is the relationship between length and area? If a meter (1m for
short) is our basic unit of distance, then the corresponding unit of area is
the square meter (or meter squared), 1m2. By definition, this is the area
of a square which is 1m long and 1m wide.
We can then use this to quantify the areas of other things. Thinking of a
1m2 square as a tile, the question is: how many tiles are needed to cover
the surface of other shapes and objects?
Of course, we don’t have to use whole tiles, they can be cut up as
necessary. A triangle which is 1m wide and 1m high occupies exactly
half of one of these square tiles, and so has an area of
For smaller shapes, we might want to use cm2 instead, meaning tiles
1cm by 1cm, while for measuring the areas of whole countries, tiles of
1km2 are more appropriate. When lengths are quoted in other units,
such as inches or miles, it makes sense to use the corresponding units for
area: square inches or square miles. It is good to be flexible about such
things. (Beware, though: the relationship between cm2, m2 and km2 is
not completely obvious! More on this later.)
The road outside my house has an area of around 500m2, meaning that
500 tiles of size 1m2 are needed. A (large) adult human being might
have a surface area of around 2m2, which tells us how much skin they
The first way to calculate (or estimate) the area of a shape is just by
counting the number of tiles needed to cover it. For instance the
rectangle on the left is covered by six tiles, while the triangle is covered
by one whole tile and two half tiles, making a total of two tiles.
Calculating, not counting
Some shapes have areas which can be calculated exactly, quickly and
easily, without the laborious task of counting tiles. My road is 100m
long and 5m wide. What this means is that 100 tiles of 1m2 would fit
along its length, and 5 tiles would fit across its width. So the road would
be completely covered by 5 rows of 100 tiles, which is why its area is
100 × 5 = 500m2. Notice that I can calculate this from just two simple
measurements, without having actually to work with tiles.
This is an example of a broader phenomenon. To calculate the area of a
rectangle, you can multiply its two distances together: area = length ×
width. A rectangle which is 8cm long and 3cm wide has area 8cm ×
3cm = 24cm2.
A square is a special case of this, because a square is nothing more than
a rectangle whose length and width are the same. So a square which is
5cm wide has an area of 5cm × 5cm = 25cm2.
Squared units versus units squared
How many cm2 are there in 1m2? The obvious answer might be 100,
since, as most people know, there are 100cm in a meter. But the obvious
answer can often be wrong! In fact, a square which is 1m × 1m is
100cm × 100cm, so its area in cm2 is 10,000cm2.
Another, similar, source of confusion is the difference between a
“hundred square kilometers” (meaning 100km2) and a “square of a
hundred kilometers” (meaning 100km × 100km). How wide is a square
whose area is 100km2? Is it 100km? No, the correct answer is 10km,
because 10 × 10 = 100. A square which is 100km long would have
area of 100km × 100km = 10,000km2.
Now, how many m2 are there in 1km2?
Beyond rectangles
The areas of rectangles are easy to calculate if we know their lengths and
widths. But other shapes are trickier. Some have their own rules for
calculating their areas. To calculate the area of a triangle, for example,
we need to multiply its height by its base and then divide by 2 (see
Triangles). This sounds easy enough, but remember the warning: the
height of a triangle may not be the length of any of its sides! The height
and the base must always be at right angles to each other.
The area of a circle is given by the famous formula πr2 (see Circles). Here
r is the circle’s radius and π is the number 3.14159… So if a clock face
has radius 5cm, then its area is π × 25cm2, which is 78.5cm2, to one
decimal place.
We use rules such as this to work out the areas of some other related
shapes. To get the area of a semicircle for example, we can calculate the
area of a circle, and then divide by 2. To calculate the area of a ring, we
can subtract the area of the inner circle from the area of the outer circle.
Beyond squares and rectangles, other quadrilaterals (four-sided shapes)
have their own rules for area. (See Polygons and solids.) The area of a
parallelogram is the same as the area of a rectangle: the length times the
height. Be careful though because, as with a triangle, the height of a
parallelogram is not the length of any of its sides, but the perpendicular
distance between opposite sides. To see why this rule should hold true,
notice that if we cut the triangular end off a parallelogram, and fit it
onto the other end, the two pieces together form a rectangle.
Volume: the science of space
Just as area measures a region in two dimensions, so volume measures
how much space objects take up in 3-dimensional space. This is useful,
since 3-dimensional space is where we live. So volume is the right way
to measure how big a physical object really is.
Just as a unit of length such as a meter gives us a way to measure area
(square meters), it also provides us with a unit of volume, the cubic meter
or meter cubed: 1m3. One cubic meter (1m3) is the volume of a cubic box
which is 1m tall, 1m wide and 1m long.
A cuboid is the 3-dimensional version of a rectangle, where six
rectangular faces meet at right angles. We can calculate the volume of
such a shape by multiplying its three dimensions. So if my room is 3m
high, 4m wide and 5m long, then its volume is 3m × 4m × 5m =
Volume is useful for measuring quantities of liquid (or gas). We often use
other units of volume for this. We might talk about the capacity of a
bottle in terms of liters, for example. There is a relationship between
liters and meters. One cubic centimeter (1cm3) is the volume of a 1cm ×
1cm × 1cm cube, and one liter is 1000cm3.
So how many liters are there in 1m3?
The volumes of solid objects
Just as with area, many different shapes come with their own rules for
calculating volume. To use these rules, we need to be able to substitute
numbers into formulae. So have a look at Algebra, if you need a refresher
on how to do that.
A sphere, for example, has a volume of
, where r is the radius of the
sphere (meaning the distance from the center to the edge). So if an
inflatable globe has a radius of 5cm, then the amount of air it contains is
(to one decimal place).
A cylinder, on the other hand has a volume given by multiplying the area
of its circular end by its height. The area of the circular end is πr2, where
r is the radius. If h is the length of the cylinder, then putting this all
together, we get a volume of πr2h.
Similarly, a cone (with a circular base) has a volume of
, where r is
the radius of the circle at the bottom and h is the cone’s height. (The
cone occupies exactly a third of the volume of a cylinder with the same
radius and height.)
So, if a wafer cone is to be filled with ice cream, and the cone is 10cm
long, with a radius at its widest end of 3cm, then the amount of ice
cream the cone can contain is
, to one decimal
place. (Of course this ignores any ice-cream sticking out beyond the end
of the cone!)
We might also need the length s, which runs diagonally from the base to
the point. (If necessary s can be calculated from r and h using
Pythagoras’ theorem, covered in a later chapter.) It turns out that the
area of the curved part of the cone is πrs.
It might be useful to put this information into a grid:
Notice that, in each case, the number of lengths multiplied together to
get the surface area is two (whether that be r2, r × h or r × s), along
with some fixed constant number (π, 2π, etc.). For the cone and the
cylinder two such terms need to be added together, to deal with the
curved parts and the flat ends.
For the volume, the number of lengths to be multiplied together is three,
whether that is r3 or r2 × h, again multiplied by some constants. So,
even among these sophisticated formulae, our golden rule holds good!
What is your volume? Unless you are perfectly spherical, cylindrical or
conical, it is unlikely that there is a neat formula to calculate the answer.
Human beings have very irregular and uneven shapes. But this is not just
true of us; most objects in the real world differ from the idealized figures
of geometers’ dreams. So how can we calculate volumes of irregular
There is a simple method, discovered by Archimedes. According to
legend, while he was having a bath, he noticed that, as he got into the
tub, the water level rose. Many people must have noticed the same
thing. But Archimedes asked himself one simple question: by how much
did the water rise?
It was then that Archimedes uttered his famous cry “Eureka!.” He
realized that the volume of bathwater displaced by his body, must be
exactly equal to his body’s volume.
To transfer the action out of the bath and into the lab, suppose we have
a measuring cup, with some water in it. The scale tells us that the
volume of the water in the cup is 100cm3. Now suppose we have some
object, whose volume we want to know. It doesn’t matter how strangely
shaped it is, or what it is made of (so long as it isn’t absorbent). If it fits
in the cup, we can measure it. When we put it in the water, the water
level rises. Let’s say it now reads 123cm3. Then we immediately know
that the volume of the object is 23cm3.
With this simple piece of equipment, we can now calculate volumes
using only subtraction. (The only thing to watch out for is that the object
is fully submerged. If part of it sticks out of the water, that part will be
missed out of the calculation.) Why not try it?
Sum up Areas and volumes both represent the size of objects, but
in different dimensions. With a bit of practice, calculating
them is not hard!
1 Calculate the area of each of the objects in parts a−d.
a A fencing strip 14m long and 2m wide
b A football pitch 100m long and 50m wide
c A computer screen 30cm wide and 20cm high
d A square photograph 4cm wide.
e There are 12 inches in a foot. How many square inches are there
in a square foot!
2 Work out these volumes.
a A fish tank is a cube, 50cm wide. How much water can it hold?
b A block of flats is a cuboid 40m high, 20m wide and 15m deep.
What is its volume?
c An award trophy is a solid gold globe with radius 4cm. How
much gold is needed to make it?
d A flotation device is a plastic cylinder that is 1 meter long and
has a circular end with a radius of 10cm. How much air does it
e A wizard’s hat is a cone, 30cm high, with a radius of 10cm at the
circular base. What is its volume?
3 What are the surface areas of the shapes in quiz 2? (For part e
you will need Pythagoras theorem. Come back to this after the
chapter on Pythagoras, if you are not yet familiar with that.)
Polygons and solids
• Knowing the different possible shapes with four sides
• Understanding how the geometry of regular polygons works
• Recognizing the Platonic solids
Geometry is the study of shapes. But what shapes are there? We have
already met the circle and the triangle. In this chapter we take a
whistle-stop tour around the wider world of geometry, to see some of
the other shapes that are out there. It’s going to be fast and furious, so
hang on to your hats! Our first stop is the quadrilateral zoo.
The quadrilateral zoo
As we saw earlier, the humble triangle comes in all sorts of different
forms: obtuse, acute, right-angled, isosceles, and so on. Quadrilaterals, or
four-sided shapes, exhibit an even greater variety. So, to begin this
chapter, we will have a nose around the quadrilateral zoo.
Its most famous inhabitant is the familiar, dependable square. This shape
is as symmetric as possible: the four sides all have the same length, and
the four angles are all right angles. Although the square itself is a very
familiar shape, a slight alteration can change its appearance. When it’s
standing on one of its corners instead of flat on its edge, it often gets
called by a different name: a diamond. But of course the two are the
same shape!
The square also serves to illustrate some of the little symbols geometers
use to communicate facts about shapes. Parallel lines (as we saw in
Angles) are marked by matching arrows. In the square, both pairs of
opposite sides are parallel. We also use matching notches to indicate
lines that are the same length. (Of course, in a square, all four sides are
the same length.) Finally, there is the little mark we use to indicate a
right angle. In a square, all four angles are right angles.
Now let’s visit one of the square’s closest cousins, the rhombus. Like a
square, a rhombus has all its four sides the same length. But, unlike the
square, its angles need not be all the same, and need not be right angles.
So a rhombus looks like a square which has been squeezed.
On the other side of the family, the square’s nearest relation is the
rectangle. Like a square, a rectangle has four right angles inside. The
difference is that the lengths of its edges are not all the same. They come
in two pairs, with opposite edges of equal length. So a rectangle has two
long edges opposite each other, and two shorter edges opposite each
A parallelogram has the same relationship to a rectangle as a rhombus
does to a square. It looks like a rectangle which been squashed, so that
its edges no longer meet at right angles. The name “parallelogram”
comes from the fact that sides which are opposite each other are always
parallel (as well as being the same length).
A trapezium (or trapezoid) is another shape which is built from a pair of
parallel lines. Unlike in a parallelogram, the remaining two sides do not
need to be parallel, or the same length. There are several subspecies of
trapezium: a right trapezium is one which contains two right angles, so
that one of its ends looks like the end of a rectangle. An isosceles
trapezium is one which is symmetric: the two non-parallel side-lines are
the same length, and meet the parallel lines at the same angles. (An
isosceles trapezium looks like an isosceles triangle with the top cut off.)
A kite is a shape which is built from two pairs of lines of equal length:
two short lines and two long lines. In that sense it is like a
parallelogram. The difference is that, in a parallelogram, the identical
lines are opposite each other and are parallel. In a kite, the identical
lines are neighbors and are not parallel.
Some of the inhabitants of the quadrilateral zoo are more exotic
creatures, what mathematicians call reflex quadrilaterals. Remember from
Angles that a reflex angle is one that is bigger than 180°. These figures
might look like a larger shape which has had a bite taken out. The most
symmetric reflex quadrilateral is an arrowhead or chevron. The definition
of this is actually the same as for the kite; it is just that the two shorter
sides meet at a reflex angle. (They go “in” instead of “out.”)
The strangest creatures in the zoo are the self-intersecting quadrilaterals.
Not everyone agrees that they should even be allowed in. These are
shapes in which two of the edges crash through each other. The most
symmetric of these are the bow-ties. A bow-tie looks like a rectangle
which has been assembled badly: one pair of edges, instead of running
parallel, crosses over.
I hope you have enjoyed your visit to the quadrilateral zoo!
Shapes with more sides: polygons
We have now investigated three-sided shapes (triangles) and four-sided
shapes (quadrilaterals). These fit into the broader category of polygons,
the general name for flat shapes built from straight lines.
As we look at polygons with more and more sides, the variety is only
going to increase. So, from now on, we will focus on only the most
symmetric shapes.
Of all the different types of triangle discussed in the chapter on triangles,
the most symmetric are the triangles where all the sides are the same
length, and all the angles are equal. These are the equilateral triangles.
Similarly, of all the quadrilaterals discussed above, the most symmetric
is the square. Again, all its sides are the same length, and all its angles
are equal.
We can continue this pattern with this chapter’s golden rule. That’s fine,
but after the equilateral triangle, and the square, what other examples
are there? There is the regular pentagon with five sides, the regular
hexagon with six sides, the regular heptagon with seven sides, and so on.
(The words here have their origin in the ancient Greek numbers.) At
each level there are also countless irregular possibilities. Just as the
square is not the only quadrilateral, so there are endless irregular
pentagons and hexagons, and so on.
Regular polygons
An equilateral triangle has angles of 60°, a square has angles of 90°.
What is the pattern here? What is the angle inside a regular pentagon,
for example?
There is a nice way to answer this question. Suppose you are walking
along the outside of a regular pentagon. You turn five corners, each the
same angle. Then, at the end, you are back facing the same way. So you
must have turned exactly 360° altogether. Each corner then, must be
. So is this the answer? Not quite!
The corner you turn when walking around the pentagon is not the
shape’s internal angle. It is the amount by which the next edge along
deviates from the line going straight on (see the diagram). This is
sometimes called the shape’s external angle, and it is this that we have
calculated to be 72°.
To find the internal angle, we need to know the relationship between
that and the external angle. But the diagram shows it: they are angles on
a straight line. That means that they must add up to 180°. So if the
external angle is 72°, then the internal angle must be 180° − 72° =
The method we have used to calculate the angle inside a regular polygon
involves two steps:
• Divide 360° by the number of sides of the polygon to find the
external angle.
• Subtract the external angle from 180° to find the internal angle.
Into the 3D world: solids
For the final stop on this geometrical mystery tour, we will take a bold
step out of the flat land of two dimensions into the world of three
dimensions. Here the counterparts of polygons are known as polyhedra,
or simply solids. The most familiar example is the cube, the big brother of
the two-dimensional square.
Polygons were two-dimensional shapes built from straight lines, which
met at corners. A solid is a three-dimensional shape built from flat faces
which meet at straight edges and corners. Other examples of solids are
cuboids, which are the three-dimensional equivalents of rectangles,
pyramids, and a host of others.
Just as we concentrated on regular polygons above, now we shall focus
our investigation on regular solids. This is the setting for one of the
greatest and most ancient of all mathematical theorems: the
classification of the Platonic solids.
The Platonic solids
Regular polygons were fairly easy to understand: we began with
equilateral triangles (three sides), then looked at the square (four sides),
the regular pentagon (five sides), regular hexagon (six sides), and so on.
For every number after that, there is a corresponding regular polygon,
and we have a very good idea of what it will look like. There are no
surprises in store!
The rules of the solid world are less clear, but the golden rule for
polygons does have a counterpart here:
A regular solid (or regular polyhedron) is one in which all the faces are
identical regular polygons, all the edges are the same length and all the
angles are the same.
That’s an elegant definition. But the big question is: apart from the cube,
what other examples are there?
This is a question which preoccupied the geometers of ancient Greece,
including the philosopher Plato. As he was able to prove, while there are
infinitely many regular polygons, the collection of regular solids is a
much smaller family. Let’s work through the reasoning that led Plato to
that conclusion.
One of the simplest regular solids, aside from the cube, is the tetrahedron.
This is a pyramid with a triangular base. Just as a cube is built from six
squares, a tetrahedron is build from four equilateral triangles (in fact its
name means “four sides”).
What are the broader rules here? The faces of any regular solid must be
regular polygons; that much is clear. So having built one from triangles,
and one from squares, a natural question is: can we build one from
pentagons? The answer is yes! Meet the dodecahedron (meaning “twelve
sides”), with its 12 pentagonal sides, which come together in threes at
every corner.
But when we try to carry on with this line of thought, we are in for a
surprise. There is no regular solid that can be built from hexagons.
Let’s see why. At every corner of a solid shape, some faces meet. There
must be at least three of them: two cannot be enough. But if we take
three hexagons, they mesh together perfectly on a flat plane. So there is
no way of fitting them together to make a shape with any depth to it.
There is no way to fit together any more than three either. (If you doubt
it, try it!)
A similar argument shows that heptagons, octagons, and so on, can
never be fitted together to make a regular solid: once you have put the
first two together, there is never enough space for a third.
So the options for building regular solids are limited to triangles, squares
and pentagons. A cube is the only solid which can be built from squares,
since the only option is for three squares to meet at each corner (as four
squares connected at a corner would lie flat). Similarly the
dodecahedron is the only possibility for pentagons. The tetrahedron is
constructed from triangles, but maybe there are other solids which can
also be built from triangles?
Yes! In a tetrahedron, three triangular faces meet at each corner. But in
an octahedron, four do. The octahedron has eight faces in total, and looks
like two square-based pyramids glued together.
There is another shape in which five triangular faces meet at each
corner: this is the icosahedron, which has 20 faces in total.
When we fit six triangles together, though, they lie flat. So we have now
exhausted every possibility.
At the end of all that we have found a family of five regular solids: the
tetrahedron, octahedron, cube, dodecahedron, and icosahedron. These
are known as the Platonic solids.
The beauty of this theorem is not only that we have found some very
pretty shapes, but that we have completed the catalog. This is a full list;
besides these five shapes there are no other regular solids!
Sum up Whether in two dimensions or in three, and whether
studying regular solids or irregular quadrilaterals, geometry is
jam-packed with fascinating facts about beautiful shapes!
1 Use a ruler to draw these quadrilaterals.
a A rectangle with sides of 2cm and 3cm
b A rhombus with sides of 1.5cm
c A parallelogram with sides of 4cm and 2cm
d A right-trapezium with parallel sides of 2cm and 5cm
e A kite with sides of 3cm and 5cm
2 Draw a picture of each of these shapes.
a an irregular pentagon (five sides)
b a right trapezium
c a regular hexagon (six sides)
d an irregular reflex quadrilateral
e an irregular ikosihenagon (21 sides)
3 Find the internal angle of each of these regular polygons.
a a regular hexagon (six sides)
b a regular heptagon (seven sides)
c a regular octagon (eight sides)
d a regular decagon (ten sides)
e a regular hexekontakaitriakosiagon (360 sides)
4 Work out the number of corners, edges and faces of each of the
Platonic solids. (Hint: in a tetrahedron, each face has three edges.
But each of these edges is shared between two faces. So the total
number of edges is 3 times the number of faces, divided by 2.)
Pythagoras’ theorem
• Understanding what Pythagoras’ theorem means
• Knowing how the theorem can be used to calculate lengths
• Recognizing how right-angled triangles are used in the wider world
Living in Greece in the sixth century BC, Pythagoras was one of the
most famous mathematicians of the ancient world. The theorem which
bears his name is one of the first great facts of geometry. Actually it is
a matter of debate whether this theorem can be attributed to him
personally rather than the group of thinkers whom he inspired, the
Pythagoreans. The theorem may even have been known to earlier
geometers in India or Egypt.
Leaving the history aside, whoever first proved it, Pythagoras’ theorem is
about triangles, those friendly three-sided creatures that we met in an
earlier chapter. Specifically, it is about right-angled triangles. These are in
many ways the most interesting triangles.
To start with, what is a right-angled triangle? Very simply, it is a triangle
which contains a right angle. That is to say, one of the three angles in
the triangle is equal to 90°. We can also see a right-angled triangle as
half of a rectangle which has been sliced in half diagonally. This
perspective will be useful later on.
Pythagoras’ theorem
The main fact we discovered in Triangles was about the angles in a
triangle. Pythagoras’ theorem describes a relationship between the
lengths of the three sides of a right-angled triangle (not any other kind of
triangle). Let’s call the three lengths a, b and c, where c represents the
longest side. In a right-angled triangle, the longest side is always the one
opposite the right angle. It even has a fancy name: the hypotenuse.
Pythagoras’ theorem describes a relationship between the squares of the
three sides, that is, the numbers you get by multiplying each length by
itself: a × a, b × b and c × c, or a2, b2 and c2, for short. (See Powers for
more discussion of squares and higher powers.)
Pythagoras tells us is that if we add the squares of the two shorter sides,
we get the square of the longest side. Putting this in algebraic terms:
a2 + b2 = c2
There is a famous geometric picture, which illustrates this fact (see
The point is that the areas of the two smaller squares (a2 and b2)
together add up to the area of the largest square (c2). But what does this
mean in practical terms—how can we use it?
Calculating with Pythagoras
How can we calculate actual lengths using Pythagoras’ theorem? Let’s
take a simple example: a right-angled triangle which is 2cm tall and 2cm
wide (so it’s a 2 × 2 square, cut in half diagonally).
What is the length of the third side (the diagonal of the square)?
Pythagoras’ theorem tells us that a2 + b2 = c2, and in this particular
example a = b = 2, and c is what we want to work out. Substituting a
= b = 2 in the formula, we get 22 + 22 = c2. Working out this out, we
find that c2 = 4 + 4 = 8.
How to we calculate c if we know c2? The answer is given by the square
root that we met in Root and logs. It must be that
. Typing this into
a calculator we get the answer: c = 2.83cm, to two decimal places.
(Notice that, even though the numbers in the question were whole
numbers, the answer isn’t. This is very common.)
Calculating the shorter sides
Sometimes we might already know the longest side of the triangle, and
want to know one of the shorter ones. For instance, suppose a rightangled triangle is 24cm wide with a hypotenuse of 25cm. The question
we want to answer is: how high is it? If we call the height a, then the
theorem tells us that:
a2 + 242 = 252
Working these squares out, this becomes:
a2 + 576 = 625
Using a technique we met in Equations, we subtract 576 from both sides,
to get:
a2 = 625 − 576 = 49
Finally, we take the square root of both sides:
, which we might
recognize (without using a calculator) as giving a = 7cm.
Distance and Pythagoras
What use is Pythagoras’ theorem? It is not as if we see right-angled
triangles every day … or do we?
Actually, it is quite common to want to know the length of a diagonal
line when we already have information about horizontal and vertical
distances. In such situations, we can get help from hidden right-angled
For example, suppose a rectangle is 5cm wide and 12cm long. How long
is its diagonal? The answer is not obvious, but Pythagoras comes to the
rescue, because when we draw in the diagonal, we split the rectangle
into two right-angled triangles that have the same length and width as
the rectangle. The diagonal of the rectangle is the hypotenuse of the
right-angled triangle. So we can use Pythagoras’ theorem. Whatever the
length of the diagonal is, call it c, it must be true that:
52 + 122 = c2
Working this out, we get c2 = 25 + 144 = 169. So we want a number
which, when squared, gives 169. With or without a calculator, we can
identify the answer as c = 13 cm.
Here is another example. Suppose a church tower is 100 meters tall. My
wife waves to me from the top, while I sit sipping coffee in a café on the
other side of the piazza below, 75 meters away from the base of the
tower. How far is my wife from me?
There is a right-angled triangle hidden in this scenario. One edge is
provided by the line from me to the base of the church tower, and
another is the church tower itself. These are 75m and 100m respectively,
and crucially these two are at right angles to each other (assuming that
the tower goes straight up and down, not like the leaning tower of Pisa).
The third side (the hypotenuse) is now the length we want, because this
has me at one end and my wife at the other. Again we call this length c,
and Pythagoras’ theorem assures us that:
752 + 1002 = c2
To solve this, we first work out 752 = 5625 and 1002 = 10,000. Adding
these together, we find that c2 = 15,625. So the final step is to take the
square root of 15,625, and find that c = 125m.
A beautiful proof!
I hope you have seen that Pythagoras’ theorem can be genuinely useful
for calculating lengths, whenever a right-angled triangle can be found.
But why should it be true? This theorem has been proved in many
different ways, probably more than any other theorem in the history of
mathematics. Each generation of mathematicians provides a new proof.
In 1907, Elisha Loomis assembled a collection of 367 different proofs,
but the total number is much higher.
One particularly neat proof was discovered by Bhaskara in 12th-century
India. Start with a right-angled triangle with sides of length a, b and c,
where c is the hypotenuse as usual. Now we form two large squares,
each with side length a + b.
The first large square is divided up inside into four copies of the original
triangle, together with one smaller square of area a2 and another square
of area b2. The second large square is divided up differently, into four
copies of the original triangle and one smaller square of area c2. The
crucial observation is that the areas of the two large squares are the
same (obviously, because they are identical). In each case, there are four
copies of the right-angled (a, b, c) triangle inside; although these are
arranged differently, the total area of these four triangles must be the
same. That means that the area that is left over in each of the two large
squares must be the same. In the first square the area left over is a2 + b2
and in the second square the area left over is c2 so these two quantities
must be equal, that is, a2 + b2 = c2.
Such a wonderful theorem is worth spending some time thinking about
—even if it’s not immediately obvious!
Pythagorean triples
Perhaps the simplest right-angled triangle is half a square. If the square
is 1cm wide, then Pythagoras’ theorem tells us that the hypotenuse c
satisfies c2 = 12 + 12 so c2 = 2, and it must be that
. It is
slightly inconvenient that this is not a whole number. In fact, it is even
worse. Like the number π (see Circles),
is an irrational number,
meaning that it cannot be written exactly as a fraction, or even as a
recurring decimal.
This is what usually happens. For example, if you draw a right-angled
triangle with shorter sides 1cm and 2cm, the hypotenuse will be
which again is irrational.
Occasionally, however, we do get a whole number as the answer. If the
two shorter sides are 3cm and 4cm, then the hypotenuse is exactly 5cm.
These situations, where all three sides of a right-angled triangle are
whole numbers, are known as Pythagorean triples. The first one is 3, 4, 5.
If we double all these lengths, we get another: 6, 8, 10. Multiplying all
the lengths by other numbers will also produce further Pythagorean
Apart from multiples of 3, 4, 5, the next Pythagorean triple is 5, 12, 13.
Pythagorean triples are rather mysterious and unpredictable things!
Sum up Pythagoras’ theorem is one of the great geometrical
theorems. It is just a question of putting it to good use!
1 Calculate the hypotenuse (the longest side) of right-angled
triangles which have these shorter sides.
a 3 miles and 4 miles
b 1cm and 1cm
c 2 meters and 3 meters
d 5mm and 6mm
e 8km and 15km
2 Calculate the missing side! In each case c is the hypotenuse, and
a and b are the shorter sides.
a a = 1mm and c = 3mm
b b = 2 miles and c = 4 miles
c a = 1cm and b = 9cm
d b = 7km and c = 11km
e a = 11 inches and c = 61 inches
3 Calculate these distances (to one decimal place). In each case,
you will need to identify a right-angled triangle, and spot which
side is the hypotenuse.
a A rectangular room is 4 meters long and 3 meters wide. What is
the distance from one corner to the diagonally opposite corner?
b A flag is 38cm wide and 56cm long. A black stripe runs
diagonally from one corner to another. How long is the black
c An airplane flies directly over my friend’s house at an altitude of
10km. I am 1km away from her house. How far is the airplane
from me?
d A football pitch is 119m long and 87m wide. If I run from one
corner to the diagonally opposite corner, how far have I run?
e A submarine is 300m east of a ship on the sea surface, and is at a
depth of 400m. How far apart are the ship and the submarine?
4 a Draw a right-angled triangle. Now try to recreate Bhaskara’s
proof using your triangle!
b A right angle triangle has sides of length a = 3cm, b = 4cm,
and c = 5cm. Find its area (see chapter on Triangles) and find
the areas of the four squares used in Bhaskara’s proof. Check that
the proof works!
c In a Bhaskara proof, the four squares have areas of 6.25cm2,
36cm2, 42.25cm2, and 72.25cm2. What are the dimensions of the
right-angled triangle?
d In a Bhaskara proof, the largest outer square has an area of
529cm2 and the smallest inner square has an area of 64cm2.
What are the dimensions of the right-angled triangle?
5 Use Pythagoras’ theorem to complete these Pythagorean triples.
a 7, 24, ?
b 8, 15, ?
c 9, 40, ?
d 11, ?, 61
e ?, 35, 37
• Understanding what trigonometry is all about
• Knowing how to use trigonometry to understand triangles
• Deciding which of sin, cos and tan to use
So far, we have seen that triangles of any shape have one thing in
common: their three angles add up to 180° (see Triangles). We have
also met the famous Pythagoras’ theorem, which relates the three
lengths of any right-angled triangle. What we don’t yet have is anything
which relates the two: something which allows us to calculate the
lengths in a triangle if we know the angles, and vice versa. This is the
subject of trigonometry, known as trig to its friends, a word which
literally means “triangle-measuring.”
Similar triangles
Let’s have a look at some right-angled triangles. Suppose we fix one of
the angles to be 60°, and we want to know what the lengths of the sides
are. Notice that we now know two of the angles: the right angle (90°)
and the angle of 60°. Since the three angles must add up to 180°, it must
be that the third and final angle is 180° − 90° − 60° = 30°.
Now, none of this determines the lengths of any of the sides. It could be
that the triangle is 100 miles long or just 0.5mm. Yet, when we look at
different triangles with this same arrangement of angles, there is
something very striking about them.
Although these triangles are different sizes, when we look at them, they
are all clearly the same shape. (The technical term for being the same
shape, but not necessarily the same size, is being similar.) The moral is: if
we know all three angles then, as soon as we fix the length of one of the
sides, in order for the triangle to have the right shape (and thus have the
correct angles), the other two sides are automatically determined. There
is no further room for maneuver. This is the key idea behind
trigonometry. In the rest of the chapter we will put some technical meat
on these bones.
Keeping things in proportion
What does it really mean for two triangles to be the same shape, but not
necessarily the same size? It means that their three angles must match.
That’s fine, but what does it mean for the lengths of the sides? Well,
however long or short they are, the three sides in the two triangles must
be in the same proportion to each other.
In the 60° triangles above, it turns out that the shortest side (the one at
the bottom) is always half the length of the longest side (in the
terminology of Pythagoras’ theorem that’s the hypotenuse). So, if we are
presented with a right-angled triangle which also contains an angle of
60°, and we know that the hypotenuse is 4cm, then it automatically
follows that the shortest side must be 2cm long.
It is customary to name the three sides of the triangle H (for hypotenuse),
A (for the side adjacent to the angle we are focusing on, in this case the
60° angle) and O (for the side opposite that angle). What the last
paragraph says is that, in any such triangle, it will always be true that
Now, the number
is specific to the angle of 60°, but the line of
reasoning is not. For any angle that can occur in a right-angled triangle,
there is a fixed number to which
will always be equal. Usually these
numbers are less neat than . In the case of 45°, for example, it will
always be true that
. In the case of 30°, it will always be true that
The word we use for this number is cosine. We say that the cosine of 60°
is , the cosine of 45° is
, and the cosine of 30° is
abbreviated to cos, and we would write:
cos(30°) =
. Cosine is usually
, and
, and
What is the relationship between the angle and the corresponding
number, its cosine? How can we calculate the cosine? Here, my advice is
simple: leave it to your calculator! This is not like addition or
multiplication, or even like logarithms: there is no simple way to
compute the cosine of an angle by hand. But your calculator does have a
button which can do it for you:
. So, to calculate the cosine of 18°,
you would type
Warning! Make sure the mode is set to degrees not radians, otherwise
these calculations will come out wrong! (You may need to consult your
calculator’s handbook to do this.)
Calculating lengths from angles
Let’s suppose that we have a right-angled triangle. We know that one of
its angles is 18°, and we know that the hypotenuse (H) is 4cm long. This
determines the length of the adjacent side. But what is that length? Let’s
calculate it. Whatever the value of cos(18°), it must be true that
. We also know that H = 4cm.
We want to know A, but this equation gives us . So we need to multiply
by 4. The rules of equations apply here, specifically the command to
keep the equation balanced by doing the same thing to both sides (see
Equations). Multiplying both sides by 4, therefore, we get A = 4 ×
This we can type straight into a calculator:
, giving
an answer of 3.8, to one decimal place. So the length of the adjacent side
is 3.8cm. Notice that it was worth resisting calculating cos(18°) itself,
until the very last moment!
The argument is slightly different if we are faced with a right-angled
triangle where we know an angle and the length of the adjacent side,
and we want to work out the hypotenuse. Suppose a right-angled
triangle has an angle of 75°, with an adjacent side of 6 meters. This
means that
, but in this case we know that A = 6m.
. We want to get H on its own on one side of the equation,
so we start by multiplying both sides by H. This gives us 6 = H ×
cos(75°). Now, cos(75°) is just a number, even if we don’t know its value
yet, so we can divide both sides by that to get
This is now something we can tap into our calculator:
, which gives an answer of 23.2 meters to one
decimal place.
So far, we have focused on the relationship between the adjacent side
(A) and the hypotenuse (H), as expressed by the cosine of the angle
between them, which we might call x. But there are similar relationships
between the other pairs of sides too. Between the opposite side (O) and
the hypotenuse (H), we have the sine, or sin for short: sin
. With
this, if we know an angle and the length of the opposite side, we can
calculate the hypotenuse H (and vice versa).
Similarly, between the opposite side (O) and the adjacent side (A) we
have the tangent, or tan for short, defined by tan
Armed with the three trigonometric functions, sin, cos and tan, it is
possible to work out a lot of information about triangles. But first we
need to remember which is which! There are various mnemonics to help
us remember these; the shortest is “SOH-CAH-TOA”:
(Soh-Cah-Toa is phony Chinese, very roughly translating as “Big Foot
Aunt” in Mandarin.)
The power of trig!
Let’s see the power of trigonometry. Suppose a tower is 30 meters away
from me. When I look at the top of the tower, the angle of elevation
from me to the top is 15°. The question is: how tall is the tower?
There’s a right-angled triangle here, in which the three corners are: the
point where I am standing, the base of the tower and the top of the
tower. So, we can use trigonometry. What is more, we know one of the
angles: 15°. The first thing to decide, then, is which trigonometric
function (cos, sin or tan) we want to use. We know the adjacent side (A):
30m. What we want to know is the opposite side (O): the height of the
tower. So, SOH-CAH-TOA tells us that the right choice is the tan function
because that’s the one that involves O and A.
The TOA formula tells us that
this into a calculator
, so O = 30 × tan(15°). Typing
, we get an answer of 8.04
meters to two decimal places.
All the examples we have seen so far fit a similar pattern: • We have a
right-angled triangle.
• We know one of its angles (apart from the right angle), as well as
the length of one of its sides.
• We want to calculate the length of one of its other sides.
• Pick the appropriate trigonometric funtions (sin, cos and tan), using
• Plug the numbers we have into the formula.
• Rearrange the equation.
• Calculate the answer.
Doing trigonometry backward!
There is another way in which trigonometry can be useful, which doesn’t
follow the above pattern. If we know the lengths of two of the sides of a
right-angled triangle, then we can calculate the remaining angles (the
ones that are not the right angle). Of course, we can also calculate the
third side, using that old standby, Pythagoras’ theorem.
For instance, suppose we know that a triangle has a hypotenuse of 5m,
and one of its other sides is 4m long. Let’s say we want to know the size
of the angle opposite the 4m side; call the angle x. SOH-CAH-TOA tells
us that sin
How we can use this to find out x? Well, what we need to do to the left
of the equation is “undo” sine, somehow, and on the basis of always
doing the same thing to both sides of the equation, we need to “undo
sine” on the right as well.
This may be starting to sound like nonsense, but actually it is perfectly
meaningful. Sine is a rule which takes in a number representing an
angle, and spits out another number (representing the ratio of two sides,
in the related right-angled triangle). There is another rule which does
the same thing backward: it takes in the value of
and spits out the
corresponding angle. This function is called inverse sine, and is written
sin−1 (or sometimes “arcsin”).
Inverse sine can be found on most calculators by pressing
followed by
. So, to complete the example above, we have arrived at
(converting the fraction to a decimal: see Fractions if you
have forgotten how to do this). Now we need to take the inverse sine of
both sides: x = sin−1(0.8). All we need to do now is calculate this using
a calculator:
, giving a final answer of 53°, to the nearest
There are also inverse cosine and inverse tangent rules, which relate to
cosine and tangent in exactly the same way.
Sum up Trigonometry is the science of measuring triangles, and
its three main planks are sin, cos and tan. So long as you have
a calculator, and can remember SOH-CAH-TOA, trigonometry
is not too hard at all!
1 Use a calculator to find these.
a cos(72°)
b cos(61°)
c cos(11°)
d cos(59°)
e cos(28°)
2 Calculate the length of the adjacent side, when the hypotenuse
(H) and the given angle (x) are as follows.
a H = 6 miles, x = 54°
b H = 3cm, x = 12°
c H = 3 meters, x = 77°
d H = 12 nanometers, x = 45°
e H = 45mm, x = 34°
3 In each case below, x is an angle in a right-angled triangle, A is
the adjacent side, O is the opposite side and H is the hypotenuse.
Choose the correct trig function, and calculate the unknown length
to one decimal place.
a x = 35°, O = 2 miles, what is H?
b x = 22°, A = 5cm, what is O?
c x = 70°, H = 1km, what is O?
d x = 41°, A = 67mm, what is H?
e x = 12°, O = 1 micrometer, what is A?
4 In each case below, the lengths of two sides of a right-angled
triangle are given. Choose the correct inverse trig function to
discover the angle that is opposite the side O and adjacent to the
side A.
a A = 3cm, H = 5cm
b H = 6mm, O = 1mm
c A = 5 miles, O = 7 miles
d O = 12km, H = 14km
e H = 55 meters, O = 24 meters
• Understanding what axes are
• Interpreting coordinates
• Knowing how to plot points on a graph
So far, we have met a variety of geometrical objects and techniques. To
take the subject further, we need a feeling for the mathematical space
where geometry takes place, the background on which our lines and
shapes are drawn and studied. In two dimensions, this role is played by
the plane, essentially an idealized piece of paper. Unlike any paper in
the physical world, however, it is perfectly smooth, flat, and—well—
plain. And it is infinite, extending forever in all directions. Of course,
no physical piece of paper can hope to achieve this! How do you know
where you are on an infinite piece of paper? The answer comes in the
form of Cartesian coordinates, named after the French philosopher
René Descartes, also known by his pen-name, Cartesius.
Coordinates are familiar to anyone who has ever read a map, and
Cartesian coordinates work in a very similar way, as a map of the
infinite plane.
Introducing the axes
The starting point for Cartesian coordinates is as follows: there are two
lines drawn across the plane. One runs horizontally from left to right,
and the other runs vertically up and down. These are known as the
horizontal and vertical axis respectively (or as the x-axis and y-axis, for
reasons we shall come to in the next chapter).
Each of the two axes has numbers along it. The horizontal axis is
identical to the number line that we met in Negative numbers and the
number line. In the middle is the number 0. To its left are the negative
numbers: −1, −2, −3, … To its right are the positive numbers: 1, 2, 3,
… (We are not only interested in the whole numbers here: between 1
and 2 is a full range of fractions and decimals.) The vertical axis, as its
name suggests, runs from down to up instead of from left to right, but is
otherwise identical to the horizontal axis. These two lines meet in only
one place: where they are both 0. This special point is known as the
This pair of axes forms a cross in the middle of the plane. But they can
be used to locate every point on the plane. This is the idea of Cartesian
Descartes’ infinite map
A pair of Cartesian coordinates consists of two numbers, usually written
inside brackets, and separated by a comma like this: (4, 5). This pair of
numbers pinpoints a position on the plane. How? These are the rules: •
The first coordinate says how far left or right the point is from the origin.
In this case it is 4, so the point will be in line with the 4 on the
horizontal axis.
• The second coordinate (5 in this case) tells us how far up or down
the point is from the origin. So it must be in line with the 5 on the
vertical axis. The most common type of confusion with coordinates
is getting is getting (4, 5) muddled up with (5, 4). You need to
remember that the first number in the pair counts from left to right,
and the second from down to up. This chapter’s golden rule is a
mnemonic for remembering the correct order of the coordinates. If
you are used to reading ordinary map references, you will find this
easy as the system is essentially the same.
We also say that the point (4, 5) has an x-coordinate equal to 4, and a ycoordinate equal to 5. (Now we can remember that the coordinates
appear in alphabetical order: we just have to remember which axis is
which!) The origin has coordinates (0, 0).
When we have two points on the plane, an obvious question to ask is
how far apart they are. How can we get this information from
The answer comes from that bedrock of geometry, Pythagoras’ theorem,
which we discussed earlier. Let’s look at the two points (−1, −2) and
(3, 1). The first step is to work out the horizontal and vertical distances
between the points.
The horizontal coordinate is −1 at the first point and 3 at the second,
which is an increase of 4. That is the horizontal distance between the
points. Similarly, the first vertical coordinate is −2 and the second is 1,
which gives a vertical difference of 3. That is the vertical distance
between the points.
These are the horizontal and vertical sides of a right-angled triangle, so
Pythagoras’ theorem tells us how to use these two figures to work out
the distance between the two points. The length we want is that of the
hypotenuse. If we call that c, then c2 = 32 + 42, which gives an answer
Breaking this down, the method for calculating the distance between two
points is as follows: • Calculate the horizontal and vertical distances
between the two points.
• Use Pythagoras’ theorem to calculate the direct distance.
Sometimes there is a short cut. Consider the two points (1, 2) and (5, 2).
In this case the horizontal distance is 4, and the vertical distance is 0. So,
in this case, the actual distance is the same as the horizontal distance: 4.
Into three dimensions!
The plane is the right setting for studying circles, triangles, and so on.
But what about the likes of spheres and cubes? There is no room in the
plane for objects like this, which have depth as well as length and
This is the moment we need to step from two dimensions into three. The
system of 3-dimensional coordinates works very similarly to the 2dimensional one. The difference is that there is an extra axis. Before we
had only an x-axis and a y-axis. These are now joined by a z-axis.
Usually the three axes are arranged as shown in the diagram.
This time, any point in 3-dimensional space can be represented by a
triplet of numbers such as (1, 2, 3), which give the point’s position along
the x-, y- and z-axes respectively. The biggest challenge here is artistic! It
helps to draw in guidelines parallel with the axes, to show where your
point really is.
Pythagoras in three dimensions
Finding the distance between two points in 3-dimensional space is
almost the same as in two. Suppose we want to find the distance
between the points (1, 2, 3) and (3, 5, 7). You might be able to guess the
method! First find the distance between the two points along each
coordinate axis. The x-distance is 3 − 1 = 2, the y-distance is 5 − 2 =
3, and the z-distance is 7 − 3 = 4. Then, following the method that we
used in two dimensions, we square each of these distances, add them up
and take the square root:
, to two decimal places.
Although this method is a straightforward adaptation of the 2dimensional version, you might wonder exactly what is going on. After
all, we seem to be using Pythagoras’ theorem, but with three numbers
being squared and added up to find a fourth. What can this mysterious
four-sided triangle be?
It is not so mysterious really! If we ignore the z-dimension for a moment,
then the distance between the two points is
, using Pythagoras’
theorem just as before. We might call this the x−y-distance.
But the x−y-distance (that’s
and the z-distance (4) form the
first two sides of another right-angled triangle, and what we want to
know is the hypotenuse. So again we square them, add them together,
and then take the square root. But squaring
32, so the answer will be
gives us back 22 +
, as expected!
Sum up Axes and coordinates are to geometers what a map and
compass are to mountaineers!
1 What are the coordinates of each of the points a to e?
2 Draw and label a pair of axes, and plot these points.
a (1, 2)
b (−2, 1)
c (3, −3)
d (−3, −2)
e (−2 , −2 ) 3 Calculate the distance between each of these pairs
of points.
a (1, 2) and (5, 5)
b (0, −4) and (5, 8)
c (−1, 1) and (1, −1)
d (−2, −2) and (−2, 1)
e (−2, −3) and (6, 7)
4 Draw a set of x-, y-and z-axes, and plot these points.
a (1, 1, 1)
b (3, 2, 1)
c (−1, 3, 2)
d (2, −2, 1)
e (0, 3, − 3)
5 Find the distance between the point (2, 2, 2) and each of the
points in quiz 4 above.
• Knowing how to plot a graph
• Finding the equation of a line
• Understanding how geometry and algebra are related
In the last chapter, we saw how to equip the plane with axes and
coordinates. This now gives us two ways of looking at it: the first comes
directly from the lines and shapes that we can draw, and the second is
numerical, coming from the coordinates of the points within those lines
and shapes. How these two perspectives are related is a very deep
question, one that mathematicians continue to study today. In this
chapter we look at the ways to move between the two, in the particular
case of straight lines.
Let’s start by looking at the collection of all the points which have
coordinates like this: (0, 0), (1, 1), (88, 88), (2, 2), (−1.5, −1.5), etc.
The numerical pattern is, I hope, clear. These are the points whose two
coordinates are equal.
But what pattern do they make when we draw them on the plane? There
is only one way to find out!
What we have here is a straight line. What more can we say about it?
Well, to start with, we could ask about its relationship with those two
other lines: the axes. Where does it cross them? The answer is that it
crosses each of them at the same place: the origin (0, 0). What else
might we want to say? On closer inspection, we can see that our new
line is at an angle of 45° to the horizontal.
From a geometric perspective, these pieces of data, that it is a straight
line, which passes through the origin, and has an angle of 45°, pin it
down uniquely. If you want to draw something fitting this description,
there is only one possible line you can draw.
The equation of a line
When we have a pair of coordinates, it is traditional to refer to the first
one as x, and the second as y, like this: (x, y). So the point (8, 9) has an
x-coordinate of 8 and a y-coordinate of 9. (This is the reason why the
horizontal axis is also known as the x-axis, and the vertical as the y-axis.)
What is the purpose of this extra jargon? Let’s return to the line that
comprises all the points like (1, 1), (88, 88) and (−1.5, −1.5). What is
the defining characteristic of this line, in numerical terms? It is that the
two coordinates x and y are always equal. We can write this as y = x.
This formula y = x is known as the equation of this particular line. The
relationship between lines and their equations is the main topic of this
From lines to equations
Let’s have another example. Look at all the points of this form: (0, 1), (1,
2), (−4, −3), (88, 89), and so on. What is the pattern here? In each case
the second coordinate (the y-coordinate) is 1 more than the first (the xcoordinate). We can write this as y = x + 1. This is the equation of this
We could equally well think of this as saying that the first coordinate is
1 less than the second, and write the equation as x = y − 1. This is
perfectly correct, but it is traditional to express the y-coordinate in terms
of the x-coordinate. So the equation of a line usually begins “y = …”
The only exceptions are lines like the one comprising all the points (4,
0), (4, 1), (4, 2), and so on. When we draw it on the graph, this comes
out has a vertical line, and it has the equation x = 4. (There is no
constraint on what value y can take: (4, 1,000,000) is another point on
this graph.)
Here is another example. Take the line comprising points like these: (0,
, (2, 1),
, and so on. The rule here is that the y-coordinate is
always half the value of the x-coordinate. So the equation is
Plotting graphs: from equations to lines
Let’s turn the whole thing on its head. Suppose we are given an
equation, such as y = 2x − 1, and we are set the challenge of drawing
the corresponding line. How can we start?
We need to find some points that lie on the line. In this context, knowing
a point means knowing its coordinates. The type of question we need to
ask is as follows: when x = 1, what is y? There is nothing magic about
the number 1 here, any other number would have done: when x = 0,
what is y? When x = 4, what is y? And so on.
The way to answer such questions is to substitute the chosen value of x
into the equation, and see what emerges for y. When x = 1, then
according to the equation, y = 2 × 1 − 1, which is 1. So we have the
coordinates of our first point on the line: (1, 1). We can draw a dot or
cross on our graph to mark this point.
When x = 0, the equation tells us that y = 2 × 0 − 1, which comes out
as y = −1. So another point on the line is (0, − 1), which we can add
to the graph.
At this moment we could stop, as two points are enough to determine
the whole line: there is only one straight line which we can draw, which
goes through both of these points. But it is good practice to calculate at
least one more point on the line, just to check that the three really do lie
in a straight line, and we have not made a mistake. (It’s always a good
plan to guard against human error.) When x = 4, the equation tells us
that y = 2 × 4 − 1 = 7. So a third point is (4, 7), which we also add to
the graph. We can finish off by connecting these three points with a line.
Steepness and gradients
So far, we have seen two perspectives on straight lines: a visual,
geometric perspective and an algebraic one, involving numbers and
If we compare the lines we have seen so far in this chapter, we might say
that one is steeper than another. This notion of steepness is critical to the
geometric viewpoint. As we shall see, it also ties in very neatly to the
equation-based approach.
First we need to make the idea of steepness precise. Earlier in the
chapter, I described a line as being at 45° to the horizontal. In doing so, I
used an angle to measure a line’s steepness. This works perfectly well
but, for the most part, we prefer a different measure, called the gradient.
When driving along a mountain road, you might have seen a sign which
says “Hill ahead 1:4” or “Hill ahead, 25%.” These expressions “1:4” and
“25%” are measures of the gradient of the hill. What they both mean, in
precise terms, is that for every 4 meters you travel horizontally, you will
travel 1 meter vertically.
This is what the gradient is: the amount by which a line (or hill) rises
vertically per unit of horizontal distance. In mathematics, this is usually
expressed as a fraction rather than a ratio or a percentage. So a geometer
would write the gradient of that hill as . The general notion of gradient
Let’s calculate this for some examples, starting with the line y = x,
which we saw above. Pick any two points on the line, say (0, 0) and (1,
1). Between these points, there is 1 unit of horizontal distance: this can
be read from the difference between the first coordinates of the two
points: from 0 to 1. What is the vertical difference between them? The
answer is the same, 1 again, as the difference between the y-coordinates
is also 1. So the gradient is
Let’s take another example: y = 2 x − 1. Pick two points on it, say (0,
−1) and (1, 1). Again the horizontal difference between them is 1 (this
is the difference in their first coordinates). The vertical difference
between them is 2, which we can see either from the picture, or by
reading from their second coordinates, since 1 − (−1) = 1 + 1 = 2.
Remember the rules of subtracting negative numbers from an earlier
chapter! So the gradient of this line is
. It makes sense that this
should have a larger gradient than the previous example, since the line
does indeed look steeper.
You might worry that the value we get for the gradient seems to depend
on the choice of the two points on the line. What if, in the last example,
we had instead picked (1, 1) and (4, 7)? This time the horizontal
difference is 3 (since 4 − 1 = 3) and the vertical difference is 6 (since 7
− 1 = 6). So the gradient comes out as
. It is no coincidence that
this is the same as the previous answer: it always will be. Unless we
mess up the calculation, the gradient does not depend on the choice of
points used to calculate it. A straight line has the same steepness,
wherever you calculate it.
A third example: the line
line are
. We saw above that two points on this
and (2, 1). The horizontal difference between these is 1
(since 2 − 1 = 1). The vertical difference is
). So the
gradient is .
Downhill all the way
In all the lines we have seen so far, as you move rightward along the
line, you also move upward. Those lines go uphill, if you like. But of
course, lines going downhill are equally possible. If we take the line y =
−x, this has points like (0, 0), (1, −1), (2, −2), and so on. The
technical term for going downhill is having a negative gradient. This
particular example has gradient of −1. How can we see this in the
numbers? Well, take the two points on the line, say (2, −2) and (3, −3).
The horizontal distance between them is 1, since 3 − 2 = 1. Notice that
we subtracted the x-coordinate of the first point from that of the second
point. Doing the same thing for the y-coordinates gives a vertical
distance of (−3) − (−2) which, remembering the rules of arithmetic
with negative numbers, comes out as −1. This negative distance may
seem a strange idea, but all it means is that the graph has lost height
over that stretch, rather than gaining it. Then applying the usual
definition of the gradient, we get a gradient of
. If the arithmetic
of negative numbers seems confusing, just remember the rule:
If the graph goes uphill it has positive gradient; if it goes downhill it has
negative gradient; if it is perfectly flat it has a gradient of 0.
Putting it all together
Let’s review the four examples above: the line y = x has gradient 1; the
line y = 2x − 1 has gradient 2; the line
has gradient ; the line y
= −x has gradient −1. There is a pattern here: the gradient always
seems to come out as the number by which x is multiplied in the
equation. This is not an illusion! So, if a line has equation y = mx + c,
for two numbers m and c, then its gradient will be m.
This is an extremely useful rule, as it means the gradient can be read
straight off the equation, with no intermediate calculations, or pictures,
The line y = 5x + 2 has gradient 5, the line
has gradient
, and so on.
A question presents itself: what do the numbers 2 and −4 signify in the
previous two equations? More generally, if a line has equation y = mx
+ c, for some numbers m and c, then it has gradient m, but what does
the number c represent?
Let’s take the line y = 5 x + 2. What it is the value of y when x = 0?
The answer is 5 × 0 + 2 = 2. Similarly, for the line
x = 0, we get
, when
. This then is the answer to the
question: the number c represents the value of y when x = 0.
In terms of the graph, this denotes the place where the line cuts the
vertical y-axis. It is usually referred to as the y-intercept.
So we have arrived at a point where we need only two pieces of data to
know the equation of a line: its gradient m, and its y-intercept c. Putting
these two numbers into the general template y = mx + c gives the line’s
This gives a way to tell whether two lines are parallel. Parallel lines have
the same gradient (equal values of m) but different y-intercepts (different
values of c). So y = 2x + 1 and y = 2x + 2 are parallel lines, for
example, while y = 3x + 1 is not parallel to those two.
This is a very useful fact! For example, the rule for converting
temperature measured in degrees Celsius into degrees Fahrenheit is to
multiply by 1.8 and then add 32. In other words, the equation of this
relationship is y = 1.8x + 32, where x is a temperature in °C and y is
the equivalent temperature in °F. We can now instantly draw a graph of
this relationship: a straight line with gradient 1.8 and y-intercept 32.
Sum up Graphs may seem confusing at first. But remember: the
equation of any line is given by inserting just two bits of data
into the template y = mx + c, namely the gradient (m) and
the y-intercept (c).
1 What are the equations of the straight lines which contain these
a (0, 2), (1, 3), (34, 36), (−5, −3)
b (0, 0), (1, 2), (2, 4), (3, 6), (4, 8)
c (0, 0), (1, −1), (2, −2), (3, −3), (4, −4)
d (0, 0), (1, 4), (2, 8), (3, 12), (4, 16)
e (0, 1), (1, 3), (2, 5), (3, 7), (4, 9)
2 Draw a pair of axes, each running from −6 to 16. Then plot the
graphs of these straight lines on it.
a y = 3x
b y = 3x − 1
c y = −x
d y = −x + 1
3 What are the gradients and y-intercepts of the lines a–e in quiz
4 Write down the equation of each of these lines.
a The line with gradient 5 and y-intercept 4
b The line with gradient −1 and y-intercept −1
c The line with gradient
and y-intercept 2
d The line with gradient −3 and y − intercept
e The line with gradient 0 and y-intercept 8
• Knowing the difference between mean, median and mode
• Coping with large piles of data
• Understanding quartiles and percentiles
“There are three kinds of lies: lies, damned lies, and statistics.” So said
the novelist Mark Twain, attributing that opinion to British Prime
Minister Benjamin Disraeli. It is certainly true that statistics can easily
be misunderstood, and abused. Indeed, they often are. But statistical
techniques also offer the only tools to navigate through the huge piles
of data which emerge the moment we try to extract information in
serious quantities from the world.
In this chapter and the next one, we will have a look at some of the
contents of a statistician’s toolbox, keeping a particular eye out for when
they may be “lying” to us! We’ll begin by looking at three different
notions of the average of a set of data.
The mean, median and mode
The mean is what most people understand by the word average. To find
the mean of five numbers, we add them all together and then divide by
5. More generally, to find the mean of a group of numbers, we total
them all up, and then divide by however many numbers there are.
The mean can be misleading. For example, suppose a group of ten
people have a mean income of $100,000 per year. That’s ten very well
off people, you might think, but it may not be so. It could be the case
that one of them is a banker on $1 million per year, and the other nine
have no income at all.
This is an example of how outliers can distort the statistics of a group.
Outliers are individual points which lie a long way away from the
general trend of the group (that’s the single millionaire in the last
example). The mean is particularly susceptible to being skewed by
outliers. The next time you hear someone talking about the “average” of
a group (by which they will usually mean the mean), try to imagine how
misleading it might be, if the bulk of the total is due to a very few
outlying individuals, like the banker in the above example.
Another famous example is the paradox of the “average number of
arms.” Suppose there are 1,000 people in a town, of which 999 have two
arms, and 1 person has just one arm. The total number of arms is 999 ×
2 + 1, which is 1,999. So the mean number of arms per person is 1,999
÷ 1,000 = 1.999. This suggests that almost everyone in town has an
“above average” number of arms!
Despite these warnings, the mean is a very useful statistical tool.
The median is an alternative to the mean. It has two advantages over
the mean: it is easier to calculate and is far less skewed by outliers. So
what is the median? Suppose I have five children whose heights
(arranged in order) are 0.7m, 1.1m, 1.3m, 1.4m and 1.8m. The median,
very simply, is the middle value, in this case 1.3m.
Notice that, even if the tallest child is replaced by a 5m-tall giraffe, the
median is unchanged: this shows that it is much less sensitive to outliers
than the mean. But this also shows that, unlike the mean, the median
fails to take all the data into account. (For example, if a team of ten
salespeople each get paid $15,000 per year, and the boss wants to know
whether she can afford to keep them all on, she needs to know the mean
amount of money they bring in, not the median. If the mean is over
$15,000, then the team is bringing in more than they cost, but if it is less
than $15,000, then the team is losing money.)
Calculating the median of a set of values is easy: first put them in order,
and then read off the middle one. The only slight complication arises
when there are an even number of values to start with. For example, the
weights of four turtles might be: 0.6kg, 4kg, 5kg and 300kg. Here there
is no “middle one” of the four. So instead we take the middle two, 4kg
and 5kg, and then take the midpoint of these two values (that is to say,
we add the two together and divide by 2). So the median in this case is
The mode While the mean and median are the most useful averages, it
is worth mentioning this third type of average. The mode of a set of
numbers is the easiest to calculate: it is simply the number that occurs
most often. Suppose I survey the houses on my road, asking how many
people live in each house. Suppose the answers are that: 1 house is
empty, 19 have 1 person living there, 23 have 2 people living there, 15
have 3 people, and 9 have 4 people. The mode is just the commonest
answer, in this case 2.
One advantage of the mode is that it works just as well with data that
are not numerical. If three people in a house have red hair, one has
black hair and another has blond hair, then it makes perfectly good
sense to say that the modal hair color in that household is red.
The mode can be useful, but it is limited. For example, if I measure the
heights of the people in my family to the nearest millimeter, it is very
unlikely that two answers will be the same. So there is no meaningful
mode. (Even if there are identical twins in my family who do have the
same height, that answer does not carry much information.) The mode is
only relevant when the number of possible responses is limited, which is
not the case with height.
It is the case in elections, though, which is why the winning candidate is
precisely the modal candidate, that is to say the one who achieves the
largest number of votes.
Mountains of data!
We have learned the basic techniques for dealing with averages. There is
just one problem: all the examples we have seen so far have involved
only small sets of data. Statistical techniques really come into their own
when large quantities of data need to be interpreted. So let’s see what
happens! The mathematics will be exactly the same as above, but the
way we write and think about it will have to be adjusted.
If we have a hundred or a thousand data points, instead of five or six,
the first change is that, instead of writing the data in a huge list, we will
make life easier by entering it in a table.
Suppose we survey the residents of an island, asking how many times
they have traveled abroad in the last year:
The table tells us that 102 people traveled abroad 0 times in the last
year, 745 had 1 trip away, and so on.
With this new presentation of the data, we want to be able to calculate
various statistics. One statistic can just be read straight off: the mode.
The commonest response to the survey is “1,” so that is the mode.
What of the median? Listing the residents of the island in order of their
number of trips, we want to know where the middle one is. Well, we
know that the total number of people is 1,979, so the middle person is
number 990. (This is calculated by adding 1 and then dividing by 2.)
Now there is a slight obstacle, in that the table does not immediately tell
us where person number 990 sits in the table.
The best way to approach this is to add a new column. First we will
rename the “number of people column” the frequency, as it tells us how
frequent each response to the survey is. Then we add a new column to
the right, called the cumulative frequency. “Cumulative” means “adding
up as we go along” (as in “accumulate”). That is exactly what the new
column contains: the frequencies added together as we move down the
The interpretation of the new column is this: 102 people have had 0
trips, 102 + 745 = 847 have had 0 or 1 trips, then 1,438 have had at
most 2 trips (0, 1 or 2 trips), and so on. The useful thing about this
column is that we can now immediately locate the median, that is to say
the 990th person.
If we pretend that each person has a number, then people 1–102 have
each traveled 0 times, people numbered 103–847 have each traveled
once, and people 848–1,438 have each traveled twice. It is this category
that contains the 990th person. So the median number of trips is 2.
Quartiles and the interquartile range
There is other valuable information that can easily be extracted from a
cumulative frequency table.
In the example above, the cumulative frequency table arranges the
people into increasing order of the number of trips they have taken.
With this done, the median is the value at the halfway point, which we
identified as 2. We can equally well ask about the values at the quarterway point, orthe three-quarter-way point. These are known as the first
and third quartiles. (The second quartile is the median.)
To calculate the first quartile, the only tricky part is to work out where
the quarter-way mark is. Once that is done, the answer can be read
straight off the cumulative frequency table.
In the example above, there are 1,979 people, and the quarter-way point
is found by adding 1 and then dividing by 4 (or equivalently multiplying
by ), which gives the 495th person. The cumulative frequency column
tells us that this sits in the row corresponding to 1 trip abroad, so the
first quartile is 1.
Similarly the three-quarter mark is found by adding 1 and then
multiplying by , which gives us the 1,485th person. This sits in the row
corresponding to 3 trips, so the third quartile is 3.
The mean, median and mode are all measures of where the center of a
set of data is. Another fact we would like to know is how spread out it is.
A common measure of this is the interquartile range:
The interquartile range is the difference between the first and third
So, in the example above, the interquartile range is 3 − 1 = 2.
Percentiles: sifting more finely
The same line of thinking which gave us quartiles also produces
percentiles, which are a finer sifting of the data. Surprisingly the exact
definition of a percentile is not fully agreed among statisticians, but in
most cases the answers come out the same. We’ll take the very simplest
The quartiles divided the data into four parts. Similarly, we can divide it
into one hundred percentiles. To calculate the 95th percentile (for
example), take the total number of data points, which is 1,978 in the
above example, and then multiply that by the decimal which
corresponds to the percentile, which is 0.95. So we get 1,978 × 0.95 =
1,879.1. Now we want to find that number in the cumulative frequency
table, and we spot that it is in the row corresponding to 4 trips. So in
this case the 95th percentile is 4. Similarly, the 99th percentile is at the
point 1,978 × 0.99 = 1,958.22, which is in the row of the table
corresponding to 5 trips abroad, giving an answer of 5.
Sum up From the mean to the 99th percentile, the statistician’s
toolbox contains many useful devices for making sense of piles
of data!
1 Find the mean of each of these sets of data.
a The weights of five people (to the nearest kg): 25kg, 42kg, 60kg,
34kg, 45kg
b The lengths of three snakes (to the nearest 0.1m): 1.7m, 0.4m,
c The number of CDs owned by four people (exact answers): 0, 28,
12, 143
d The loudness of six dogs’ barks (to the nearest decibel): 21db,
46db, 19db, 35db, 51db, 27db
e The number of televisions in each house in street: 2 houses have
no TV, 16 houses have 1 TV, 8 have 2, and 4 have 3.
2 For each set of data in quiz 1, find the median.
3 Write a cumulative frequency column for this table. Then find
the median and mode.
The number of pets per household in a village:
4 Find the quartiles, the interquartile range and the 99th percentile
for the data in quiz 4.
• Knowing how to use numbers to measure likelihood
• Analyzing combinations of events
• Understanding how different events can affect each other
There are many aspects of the world that can be measured with
numbers. This is what makes mathematics is so endlessly fascinating!
But this is not limited to things that we can weigh or measure. Some
applications of numbers are subtler, and more indirect. One important
area is the study of probability, where we use numbers to represent the
likelihood of certain events taking place.
When we talk about an event being “likely” to happen, or “certain” to
happen, we are using the language of probability. In the study of
probability, we assign a number to this likelihood to quantify the chance
of the event happening. We use only the numbers between 0 and 1: an
impossible event has probability 0, while a certainty has a probability of
1. Everything else falls somewhere in between. For example, if I toss a
coin, then the probability of it landing on heads is
(one chance in two),
so long as the coin is fair.
Biased coins will have other probabilities. To take an extreme example, a
double-headed coin (a coin with a head on each side) will have
probability 1 of landing on heads (it is certain to happen). For the rest of
this chapter we will make a standing assumption that all the coins (and
dice and decks of cards) we meet are fair.
At one end of the scale, unlikely events have very small probabilities,
meaning numbers close to 0. The chance of your ticket winning the UK
National Lottery or the Washington State Lottery are each
(around 0.00000007). Events which are completely impossible have a
probability of 0. (As lotteries like to advertise: if you don’t have a ticket,
your chance of winning is absolutely nil!)
At the other end of the scale, very likely events have high probabilities,
meaning numbers close to 1. The probability that the sun will rise
tomorrow is very close indeed to 1, something like 0.9999…999. (I
wouldn’t want to guess the number of 9s, but in the 18th century the
naturalist George-Louis Leclerc made a serious attempt to estimate it!)
If I am asked to give a rough and ready estimate of how likely it is to
rain tomorrow, I might start by reasoning that, at this time of year, it
typically rains in my town around one day in two. This would put the
chance of it raining tomorrow at around 0.5. If it has been raining across
the entire region for several days and shows no signs of clearing up, I
might increase that estimate to, say, 0.8.
Counting successes
It is all very well estimating the probabilities of events according to how
likely they seem. But how can we work out exact answers? One basic
technique amounts to counting the outcomes of an experiment.
When we roll a standard die, there are six possible outcomes (1, 2, 3, 4,
5, 6). Suppose I want to know the probability of rolling a 5. Just one of
the six outcomes counts as a “success,” which gives us our answer: a
probability of . The rule here is that the total number of possible
outcomes goes on the bottom of the fraction, and the number of
“successful” outcomes goes on the top:
This is the basic idea. But as usual there is some fine print to take
account of! If we go back to the question of whether or not the sun will
rise tomorrow, then there are two possible outcomes: either it will or it
won’t. Of these, just one (sunrise), is classed as a “success.” So,
according the rule above, the answer should be .
The trouble with this is obvious! It is just nonsense. Sunrise is a near
certainty, and so should have a probability very close to 1.
So what has gone wrong? Well, when counting up successes and
outcomes, there is an additional rule: that all the possible outcomes must
be equally likely. This is what fails in the case of the sun. So, for the
formula to work, the dice and coins used must be fair.
Combining events: “and”
What is the point of assigning numbers to the probabilities of events? It
is not just because mathematicians are fixated with measuring
everything numerically. One benefit is that different ways of combining
events correspond very neatly to various arithmetical tricks with their
probabilities. There are two principal cases of this, which are described
by the two English words “and” and “or.”
Let us take “and” first. Suppose I roll a die and flip a coin. What is the
probability that I will roll a 6 and flip a head? We know that the
probability of rolling a 6 is , and the probability of getting a head is .
How do we mix these numbers, to get the probability for the combined
event of a head and a 6. The answer is to multiply. So the probability we
want is
, which comes out as
The general rule here is that “and” in a combined event means
“multiply” the probabilities. But we cannot just apply this rule blindly;
again there is some fine print to take into account. What, for example, is
the probability that, when I flip a coin once, I get both a head and a tail?
The answer should be 0, since that event is completely impossible. But if
we apply the rule above, without thinking about what it means, we get
an answer of
What is the caveat we need to eliminate nonsense like this, and leave us
with a rule that makes sense? The answer is that the two events whose
probabilities we are multiplying must not affect each other. In technical
terms, they must be independent. The first example passes this test: when
I roll a die and toss a coin, whether or not I get a head has no impact on
whether or not I get a 6. But, in the second example, with just one coin,
whether or not I get a head makes a huge difference to the likelihood of
my getting a tail. (In fact the one determines the other entirely.)
So we can express the rule more accurately: when two events are
independent, “and” means “multiply.”
Combining events: “or”
Let’s move on to the other principal way that two events can be
combined: “or.” When I roll a die, I might be interested in the
probability of my getting a 5 or a 6. Here we can move directly to the
method of counting up successes and outcomes, which will quickly give
us the answer:
(which can be simplified to : see Fractions). But it is
useful to think about how this answer is related to the individual
probabilities of the two separate events: getting a 5 or getting a 6. Each
of these has probability . The probability of the combined event, a 5 or
a 6, comes from adding these two together.
So the general rule is, when finding the probability of a combined event:
“or” means “add.”
Let’s have another example: Suppose I pick a card from a deck. The
probability of getting a heart is . The probability of getting the queen of
spades is
. So what is the probability of getting a heart or the queen of
spades? Well we can apply the simple rule—“or” means “add”—to get an
answer of
which comes out as
once the fractions have been
added and simplifed. (Try working that through yourself!)
As ever, though, caution is needed because this rule also comes with
some fine print. Here is why: Suppose I flip two coins. What is the
probability that I will get two heads? If I unthinkingly apply this rule, I
would reason as follows: The probability that I get a head on the first
coin is . The probability that I get a head on the second coin is
“Or” means “add,” so the probability that I get a head on the first coin or
the second coin is
. This suggests that it is a certainty. But of
course this is nonsense: it is perfectly possible that I will get two tails.
The fine print in this case is that you can only add together the
probabilities of two events when they cannot both occur. When I roll a
die, I cannot get both a 5 and a 6. So it is safe to add together those
probabilities. But I can get heads on two coins, so I am not allowed just
to add together those probabilities. In the jargon, the two events must be
mutually exclusive. This means that if one happens, then the other
If I roll one die, the two outcomes (a 5 and a 6) are mutually exclusive.
But if I flip two coins, the two outcomes (a head and a head) are not
mutually exclusive.
Now we can express the rule more accurately: when two events are
mutually exclusive, “or” means “add.”
Sum up Whenever you think something is “impossible,”
“unlikely” or “certain,” you are using the language of
probability. It has techniques to assess the likelihood of
different events happening—a valuable prize in this uncertain
1 Guess approximate probabilities for the following events
(answers may vary from person to person!).
a The next person you meet will be male.
b An asteroid will hit your house tomorrow.
c If you turn on the TV, the first person you see will be wearing
d If you pick a word on this page at random, it will have an “e” in
e Your favorite sports team will win their next match.
2 Calculate these probabilities by adding up the total number of
successes and outcomes.
a You pick a playing card from an ordinary deck. What is the
chance of getting an ace?
b You roll an ordinary die. What is the chance of getting an even
c You roll a 12-sided die. What is the chance of getting an 8 or
d You pick a card from an ordinary deck. What is the chance of
getting a spade?
e You roll a 20-sided die. What is the chance of getting a prime
3 Are these pairs of events independent?
a You toss a 10c coin and a 25c coin and get a head on the 10c
and a tail on the 25c coin.
b You roll a die and pick a card from a deck. You get an even
number on the die and a king.
c You pick a card from a deck, replace it, shuffle, and pick another
card. Both times you get an ace.
d You pick a card from a deck, don’t replace it, and then pick
another. Both times you get an ace.
e You pick a single card from a deck and get a black card and an
4 In quiz 3 above, where the pairs of events are independent,
calculate the combined probability of both occurring.
5 Are these pairs of events mutually exclusive? Where the answer
is yes, calculate the probability of one or the other happening.
a You pick a card from a deck and get the queen of spades or a
b You roll a die and get an odd number or a 6.
c You toss a 10c coin and a 25c coin and get a head on the 10c
coin or a head on the 25c coin.
d You roll two ordinary dice, and their total is 2 or their total is
e You pick a card from the deck, replace it, shuffle and pick again.
You get the ace of spades twice, or the queen of hearts twice.
• Understanding how to interpret pie charts and bar charts
• Representing proportion graphically
• Knowing how to convert raw data to charts
Every day, in every newspaper, magazine and current-affairs website,
you will find a wealth of statistics. Often, though, these numbers are
not displayed in tables or lists but are incorporated into diagrams of
the statistics. There are several different types of diagram or chart, and
most are easy to understand visually. Indeed, this is why they are used!
In this chapter, we will have a look at these charts in more depth; you
will see how to interpret them and how to understand the rules for
producing charts yourself. Most spreadsheet programs have tools for
creating such charts, and if you want nice-looking charts for a
presentation, that is the best way to proceed. You can think of this
chapter as a behind-the-scenes glimpse of what these programs do.
Pie charts
The first type of diagram we will look at is the pie chart. The name is
“pie” as in something delicious baked in the oven, rather than the
number π that we met in Circles. But, as it happens, both are relevant,
since a pie chart is essentially a circle, divided up into differently
colored slices. The idea is that the size of each slice corresponds to some
proportion of the whole.
Let’s take an example. In my city, one third of the people have blond
hair, one third brown and one third black. To represent this information
in a pie chart is straightforward: first draw a circle, and then divide it
into three slices. One slice represents the people with blond hair, another
those with brown hair, and the third those with black hair. Crucially, in
this case, the three slices must be the same size, because the three
sections of the population are the same size.
Now we come to the geometric nub of the matter. How do we divide a
circle into thirds? It is easy enough to do it approximately by eye, but
we want to do it exactly. The slices will all meet at the center of the
circle. The key to the matter is the angle of each slice. Because the three
slices are all intended to have equal size, the three angles must be equal
too. What is more, in the terminology of the chapter Angles, these form
angles at a point, which means that they must add up to 360°. So we are
looking for three equal angles which add to 360°. It is obvious, I hope,
that each one must be 360 ÷ 3 = 120°. Once the angles have been
established, it is just a matter of drawing the chart, using a protractor.
Of course, things are trickier when the proportions are not all the same.
In the next town along, half the people have brown hair, three eighths
have black hair and just one eighth have blond hair. How can this be
represented as a pie chart? The clue is in the proportions: one half, three
eighths and one eighth. All we need to do is split up the angle at the
center of the pie according to these proportions. This amounts to
multiplying 360° by each of the proportions in turn. To start with,
. This represents the largest slice: the half of people who
have brown hair. (It should not be a surprise that 180° looks like a
straight line.) Next,
. (You can calculate this by
multiplying 360 by 3 and then dividing by 8.) Finally,
. With
these angles, it is now easy enough to draw the chart.
From raw data to proportions
If we want to draw a pie chart, we need to be able to calculate
proportions from the raw statistics, rather than having the proportions
given to us. Usually, things will not come out as neatly as
or a ! But
precisely the same line of thinking as above will yield a beautiful pie
chart, even for messier sets of data.
Suppose, in my town, there are 1,794 people who are right-handed and
215 who are left-handed. To express this data in a pie chart, we need to
know what angles correspond to the slices for “right-handed” and “lefthanded.” To do that, we first need to know the proportion of righthanded and left-handed people in the whole population.
The total number of people is 1794 + 215 = 2009. So the proportion of
right-handed people is
, and that of left-handed people is
. We
could convert these to decimals or percentages, but there is no need.
Instead, we can now work out the angles for the pie chart. The angle
corresponding to right-handed people is now
, which we
calculate by multiplying 360 by 1,794, and then dividing by 2,009, to
give an answer of 321.5° (to one decimal place). Similarly
(to one decimal place). These are the angles needed for
the pie chart, which can then be drawn easily.
Bar charts
Even more familiar than pie charts are bar charts. The starting point this
time is a pair of axes, similar to those we used in Coordinates. The
vertical axis has the scale on it. So if we are measuring the populations
of countries, then the vertical axis might be labeled with numbers such
as 10 million, 20 million, 30 million, and so on.
The horizontal axis is the ground from which bars grow. The first bar,
for example, might represent the UK: the height of the bar carries the
data. In this case, it represents the population of the UK, around 63
million. Then the next bar represents the next country, and so on. It is
good practice to keep the bars equally spaced.
(Sometimes you see bar charts with the axes switched round, so that the
bars extend from left to right, as if running a race, rather than growing
from the ground, like buildings.)
The idea of a bar chart is simple enough: this is why they are so
commonly used. The main element of skill is in choosing the right scale
for the data. For example, if I wanted to include China (population 1.3
billion) and India (1.2 billion) in the above chart, then the scale I used
above is not suitable, as the bars for China and India would be too long
to fit on the page. It would be better to have a scale in billions this time.
Of course this makes the UK bar very small, but that’s unavoidable.
On the other hand, if I was comparing the populations of various small
islands such as Fair Isle (population 72) and Bressay (population 390),
then a scale which increased in hundreds would be better.
So, the first step to creating a bar chart is to look at the numbers
involved, and choose a sensible scale. Pick a maximum number: ideally a
nice round number, which is slightly more than the biggest number you
need to represent on the chart. I might pick 500 for the islands example
above. After that, all that remains is to draw the axes, write in the scale
on the vertical axis (making sure that the numbers increase in equal
steps, not in uneven jumps), then draw in bars with the correct heights,
and label the bars so we know what’s what.
Segmented bar charts
There are various ways in which bar charts can be spiced up to represent
subtler forms of data. One such is the segmented bar chart.
What is the purpose of it? Well, in some ways it combines the strengths
of a pie chart and a bar chart. Suppose we want to make a chart
representing the population of a certain country, and how this has
evolved over the years 2002–2011. A bar chart seems a good choice
here, with ten bars representing the ten years, and the heights of the
columns representing the population that year.
We might also be interested in what proportion of population is in fulltime work, for example. If we were only interested in those proportions,
then a pie chart would be a better choice. We could have a sequence of
ten pie charts, one for each year. Each pie chart would be cut into two
slices: one representing people who are in full-time work, and the other
representing those who are not, making it easy to compare the
proportions. Notice, though, that these pie charts contain no information
about actual numbers of people. As our golden rule tells us, they are
solely about proportions of the population, not about the size of the
We might want both of these types of data in our chart: absolute
numbers and proportions. In this case, the best option is to take the bar
chart and slightly amend it. Each bar represents the entire population in
one year. The idea is to split it into two segments: one representing the
people in full-time work, and the other those who are not. Now we can
easily compare the size of the full population from year to year, as well
as the numbers of people in full-time work.
Sum up There are many different ways to represent data: bar
charts, pie charts and segmented bar charts are just a few. All
of them are easy to understand visually—after all, that is the
1 Calculate the angles needed for these pie chart.
a In a block of flats, a quarter of the flats have one resident, half
have two, and a quarter have three or more.
b On a menu, two thirds of the dishes are vegetarian, one sixth
contain meat, and one sixth contain fish.
c One day, a TV channel dedicates one fifth of the time to news,
three fifths of the time to drama, one tenth of the time to music,
and one tenth to advertising.
d In a bookshop, five eighths of the books are fiction, one eighth
are reference, one eighth are biography and one eighth are
assorted non-fiction.
e In my house, seven tenths of the wall-space is painted, one fifth
is wallpapered, and one tenth is tiled.
2 Draw a pie chart for each of the scenarios in question 1.
3 Calculate the proportions, and angles, needed for the pie charts
in these situations.
a A room contains 10 men and 12 women.
b A radio station dedicates 1 hour a day to news, 1 hour to
advertising and the rest to music.
c One month, a cinema shows 8 thrillers, 15 comedies, 2 horror
films and 5 children’s films.
d A man’s CD collection contains 124 rock albums, 17 jazz albums,
36 classical albums and 3 spoken word albums
e A page of text contains 104 nouns, 76 verbs, 25 adjectives and
18 adverbs.
4 Draw a bar charts to represent each of these situations.
a In a village, 34 households have only cats, 75 have only dogs, 17
have both cats and dogs, while 88 have neither cats nor dogs.
b A class of students take a test. Their marks out of 5 are:
c An orchestra contains 27 string players, 14 wind players, 8 brass
players and 3 percussionists.
d A jungle contains 14.5 million herbivorous animals, 1.6
carnivorous animals and 10.1 million trees.
e A shop tracks its number of customers: in 2009 there were
10,225, in 2010 there were 12,987, in 2011 there were 15,011
and in 2012 there were 14,991.
5 Draw segmented bar charts for these sets of data. (Notice that
sometimes the first type of data includes the second, and
sometimes the two are separate.)
a The number of people in a village:
b The number of crimes reported to police:
Answers to selected quizzes
The language of mathematics page 4: Quiz 1, a 10 + 11 = 21 (true),
b 2 × 2 = 2 + 2 (true), c 5 − 4 = 2 ÷ 2 (true), d 5 ÷ 2 ≥ 3 (false), e
5 × 4 <3 × 7 (true). Quiz 2, a (1 + 2) + 3 = 6 & 1 + (2 + 3) = 6,
b (4 + 6) ÷ 2 = 5 & 4 + (6 ÷ 2) = 7, c (2 × 3) × 4 = 24 & 2 × (3
× 4) = 24, d (20 − 6) × 3 = 42 & 20 − (6 × 3) = 2, e (2 × 3) + (4
× 5) = 26 & 2 × (3 + 4) × 5 = 70. Quiz 3, a Both correct, b 4 + (6
÷ 2) = 7, c Both correct, d 20 − (6 × 3) = 2, e (2 × 3) + (4 × 5) =
26. Quiz 4, − & ÷
Addition page 11: Quiz 1, a 11, b 13, c 18, d 12, e 21. Quiz 2, a 70, b
7000, c 1100, d 11,000, e 120,000. Quiz 3, a 78, b 99, c 698, d 785, e
9775. Quiz 4, a 41, b 74, c 161, d 471, e 401. Quiz 5, a 59, b 87, c
101, d 93, e 161. Quiz 6, a 83, b 89, c 86, d 97, e 144.
Subtraction page 18: Quiz 1, a 3, b 4, c 7, d 8, e 8. Quiz 2, a 21, b 28,
c 430, d 2838, e 2140. Quiz 3, a 54, b 17, c 34, d 19, e 99. Quiz 4, a
37, b 61, c 22, d 49, e 58. Quiz 5, a 38, b 55, c 26, d 38, e 42.
Multiplication page 24: Quiz 1, a 16, b 30, c 54, d 49, e 56. Quiz 2, a
68, b 88, c 93, d 128, e 205. Quiz 3, a 880, b 690, c 480, d 1890, e
3550. Quiz 4, a 714, b 1530, c 2790, d 8733, e 54,864. Quiz 5, a 912,
b 2074, c 1653, d 11,096, e 26,292.
Division page 35: Quiz 1, a 3, b 6, c 9, d 8, e 7. Quiz 2, a 2 r 3, b 2 r 4,
c 3 r 3, d 9 r 3, e 7 r 3. Quiz 3, a 12, b 22, c 31, d 62, e 1061. Quiz 4, a
432, b 110, c 301, d 4241, e 3012. Quiz 5, a 121, b 142, c 131, d 142,
e 124. Quiz 6, a 21, b 18, c 31, d 105, e 61.
Primes, factors and multiples page 44: Quiz 1, a 3 × 5, b 2 × 3 × 3,
c 3 × 7, d 2 × 2 × 2 × 3, e 2 × 2 × 2 × 2 × 2. Quiz 2, a True &
false, b False & false, c False & false, d True & true, e True & true. Quiz
3, a Divisible by 2, 4, & 8, b Divisible by 2, 3, 6, & 7, c Divisible by 3 &
5, d Divisible by 2, 4, 8, & 11, e Divisible by 2 & 4. Quiz 4, a 2 × 3 ×
5, b 2 × 3 × 5 × 7, c 2 × 2 × 3 × 3 × 3, d 3 × 3 × 3 × 7, e 3 × 7
× 7 × 11. Quiz 5, a 3 + 7 or 5 + 5, b 5 + 7, c 3 + 11 or 7 + 7, d 3
+ 13 or 5 + 11, e 5 + 13 or 7 + 11.
Negative numbers and the number line p 50: Quiz 2, a 15 & 1, b 6 &
0, c 9 & −3, d −1 & −9, e 1 & −5. Quiz 3, a 1 & 9, b −1 & 5, c −5 &
5, d −6 & −2, e −8 & 2. Quiz 4, a −6 & 6, b −20 & 20, c −21 & 21,
d −32 & 32, e −100 & 100. Quiz 5, a 4 & −4, b −3 & 3, c 4 & −4, d
−11 & 11, e −3 & 3.
Decimals page 57: Quiz 1, a 5.5, b 13.1, c 15.50, d 10.11, e 0.01111.
Quiz 2, a 4.3, b 4.12, c 7.9, d 4.19, e 9.83. Quiz 3, a 0.8, b 0.08, c 3.5,
d 0.35, e 0.035. Quiz 4, a 3.91, b 32.24, c 9.86, d 174.76, e 45.188.
Quiz 5, a 5.3, b 0.16, c 0.2, d 10.0, e 0.720.
Fractions page 66: Quiz 1,
Quiz 3,
. Quiz 4,
, c 0.1875, d 0.285714,
. Quiz 2,
. Quiz 5, a 0.8,
. Quiz 6, All numbers except multiples of
2 & 5.
Arithmetic with fractions p73: Quiz 1,
. Quiz 5,
. Quiz 3,
or ,
. Quiz 4,
. Quiz
Powers page 80: Quiz 1, a 27, b 36, c 125, d 81, e 216. Quiz 2, a 3125,
b 46,656, c 20,736, d 15,625, e 5,584,059,449. Quiz 4,
. Quiz 5, a 222, b 215, c 155, d 1326, e 10019.
The power of 10 p 86: Quiz 1, a 7,000,000, b 8,000,000,000, c
9,000,000,000,000, d 10,000,000,000,000,000, e
11,000,000,000,000,000,000. Quiz 2, a 18 kilometers, b 37 megapixels,
c 3 nanograms, d 900 kilonewtons or 0.9 meganewtons, e 8 gigabytes.
Quiz 3, a one thousandth, b one hundredth, c one hundred thousandth,
d one ten millionth, e one trillionth. Quiz 4, a 600,000, b 21000, c
0.00000879, d 0.001332, e 67,100,000,000. Quiz 5, a 8 × 105, b 5.6 ×
104, c 6.2 × 10−4, d 9.87 × 108, e 1.11 × 10−9.
Roots and logs p 95: Quiz 1, a 2, b 10, c 8, d 7, e 12. Quiz 2, a 3.16, b
2.15, c 1.78, d 2.24, e 1.16. Quiz 3, a 11, b 2, c 8, d 125, e 81. Quiz 4,
a 2, b 4, c 3, d 7, e 3. Quiz 5, a log448, b log326, c log1072, d log542, e
Percentages and proportions p 102: Quiz 1, a 3, b 308, c 392, d
118,404, e 86.4 billion. Quiz 2, a 32%, b 16%, c 43%, d 16%, e 6%.
Quiz 3,
, b 1 & 1,
, e 0.04
Quiz 4, a 5%
increase, b 23% increase, c 32% increase, d 18% decrease, e 142%
increase. Quiz 5,
water, b Jog 2 miles & Walk 1 mile, c
50g eggs & 150g butter & 300g flour, d 40ml cordial & 320ml water, e
200g eggs & 80g onions & 160g potatoes. Quiz 6, a $231.85, b $90.05, c
$18.58, d $2806.79.
Algebra page 110: Quiz 1, a a = 2s,
. Quiz 2, a 8, b 2,
, d p = 4d + w,
, d 14, e 72. Quiz 3, a 4a, b 3b + 1, c 3x
+ 2y, d 6x − 3a, e x + 5z + 2y + 2. Quiz 4, a 4x + 4z, b 2x + 8, c
x2 −x, d x2 −2xy, e 2x2 −4xy.
Equations page 117: Quiz 1, a 6, b 7, c 6, d 8, e 28. Quiz 3, a x = 3, b
x = 12, c x = 3, d x = 2, e x = 7. Quiz 4, a x = 2, b x = 5, c x = 8,
d x = 0, e x = 3. Quiz 5, a x < 3, b x < 12, c x < 3, d x < 2, e x <
Angles page 124: Quiz 1,
b 1,
. Quiz 2, a Acute, b Acute, c
Acute, d Obtuse, e Reflex. Quiz 3, a 285°, b 210°, c 120°, d 37°, e 34°.
Quiz 4, a 90°, 90° & 90°, b 135°, 45° & 135°, c 159°, 21° & 159°, d 58°,
122° & 58°, e 4°, 176° & 4°.
Triangles page 132: Quiz 3, a 60°, b 15°, c 40° at A & 70° at C, d 55° at
A & 55° at C. Quiz 4, a 1cm2, b 2cm2, c 6cm2, d 2.5cm2, e 9cm2.
Circles page 140: Quiz 1 Exact answers (to 1 decimal place), a 12.6cm,
b 6.3cm, c 25.1cm, d 12.6cm, e 6.3cm. Quiz 2, a 15.71cm, b 31.42cm, c
3.50 miles, d 15.92km, e 0.70mm. Quiz 3, a 78.54cm2, b 19.63cm2, c
14.25mm2, d 3.56mm2, e 0.36mm2. Quiz 4 Each to 1 decimal place: a
1.3 miles, b 4.1mm, c 8.2km, d 10.2cm2, e 7.5m. Quiz 5 Each to 1
decimal place: a 28.3cm2 & 50.3cm2, b 22.0cm2, c 25cm2 & 19.6cm2, d
5.4cm2, e 77.1 cm2.
Area and volume p 148: Quiz 1, a 28m2, b 5000m2, c 600cm2, d
16cm2, e 144.Quiz 2, a 125,000cm3, b 12,000m3, c 268.1cm3 (to 1
d.p.), d 31,415.93cm3 or 0.03m3 (to 2 d.p.), e 3141.6cm3 (to 1 d.p.).
Quiz 3, a 15,000cm2 (including the top face), b 3400m2 (including the
bottom face), c 201.1cm2 (to 1.d.p.), d 6911.50cm2 or 0.69m2, e
993.5cm2 (Curved surface only).
Polygons and solids p 156: Quiz 3 (To nearest degree), a 120°, b 129°,
c 135°, d 144°, e 179°. Quiz 4, Tetrahedron: 4 corners, 6 edges, 4 faces,
Cube: 8 corners, 12 edges, 6 faces, Octahdron: 6 corners, 12 edges, 8
faces, docecahedron: 20 corners, 30 edges, 12 faces, Icosahedron: 12
corners, 30 edges, 20 faces.
Pythagoras’ theorem p 166: Quiz 1 (Each to 1 decimal place), a 5
miles, b 1.4cm, c 3.6m, d 7.8mm, e 17km. Quiz 2 (Each to 1 decimal
place), a 2.8mm, b 3.5 miles, c 9.1cm, d 8.5km, e 60 inches. Quiz 3
(Each to 2 decimal places), a 5 meters, b 67.68cm, c 10.05km, d
147.41m, e 500m. Quiz 4 b 6cm2, 49cm2, 9cm2, 16cm2, 25cm2, c
2.5cm, 6cm, 6.5cm, d 8cm, 15cm, 17cm Quiz 5, a 25, b 17, c 41, d 60,
e 12.
Trigonometry p 174: Quiz 1 (Each to 2 decimal places), a 0.31, b 0.48,
c 0.98, d 0.52, e 0.88. Quiz 2 (Each to 2 decimal places), a 3.53 miles, b
2.93cm, c 0.67 meters, d 8.49 nm, e 37.31mm. Quiz 3, a 3.5 miles, b
2.0cm, c 0.9km, d 88.8mm, e 4.7µm. Quiz 4 (Each to the nearest
degree), a 53°, b 10°, c 54°, d 59°, e 26°.
Coordinates p 182: Quiz 1, a (1, 3), b (−2, 3), c (−3, 2), d (−2, −3),
e (4, −3). Quiz 3, a 5, b 13, c 2.83 (to 2.d.p.), d 3, e 12.81 (to 2.d.p.).
Quiz 5 (Each to 2.d.p.), a 1.73, b 1.41, c 3.16, d 4.12, e 5.48.
Graphs page 188: Quiz 1, a y = x + 2, b y = 2x, c y = −x, d y = 4x,
e y = 2x + 1.Quiz 3, a m = 3 & c = 0, b m = 3 & c = −1, c m =
−1 & c = 0, d m = −1 & c = 1,
y = −x−1,
. Quiz 4, a y = 5x + 4, b
Statistics p 197: Quiz 1, a 41kg, b 1.0m, c 45.75, d 33db, e 1.46. Quiz
2, a 42kg, b 0.9m, c 20, d 31db, e 1. Quiz 3, The mode is 1 & the
median is 2. Quiz 4, The first quartile is 1, the third quartile is 3. The
interquartile range is 2. The 99th percentile is 5.
Probability p 205: Quiz 2,
Yes, d No, e Yes. Quiz 4,
. Quiz 3, a Yes, b Yes, c
. Quiz 5, a Yes, , b Yes, , c
Charts p 212: Quiz 1, a 90°, 180° & 90°, b 240°, 60° & 60°, c 72°, 216°,
36° & 36°, d 225°, 45°, 45° & 45°, e 252°, 72° & 36°. Quiz 3 (Each to the
nearest degree), a 164° & 196°, b 15°, 15°, & 330°, c 96°, 180°, 24° & 60°,
d 248°, 34°, 72° & 6°, e 168°, 123°, 40° & 29°.
acute angles 126
acute triangles 134
addition 11–17
carrying 13–14, 15
and decimals 59–60
and fractions 74–6
quizzes 17
rounding up and cutting down 16
splitting numbers in your head 15
totaling columns 13
algebra 110–16
brackets 114–15
equation solving 117–23
formulae 111–12
quizzes 116
simplifying 113–14
and substitution 112–13
alternate angles 130, 135
angles 124–31
acute 126
alternate 130, 135
at a point 128–9, 214
corresponding 130
measuring and drawing 126–8
obtuse 126
opposite 129
and parallel lines 129–30
and pie charts 213–15
quizzes 131
reflex 126, 159
in a regular polygon 160–1
right 126
on a straight line 128–9, 135
translating between degrees and revolutions 126
in triangles 133, 134–6
types of 126
using a protractor 127–8
Archimedes 154
area 149–51
calculating 150–1
circles 144–5, 151
parallelograms 151
quizzes 139, 146, 155
rectangles 138, 150
squared units versus units squared 150
squares 144, 149, 150
triangles 136–8, 151
arrowhead (chevron) 159
averages 198–200
axes 183–6
bar charts 215–16
segmented 216–18
BEDMAS 9, 120
billion 88
bow-ties 159
brackets 8–9
and algebra 114–15
cosine 177
logarithms 98
π (pi) 143
powers 81
roots/square roots 96, 97
canceling 76–7
Cartesian coordinates 183–6
Celsius, converting into Fahrenheit 195
centimeters 91
charts 212–19
bar 215–16
pie 213–15
quizzes 218–19
segmented bar 216–18
chunking 37–8
circles 140–7
area 144–5, 151
circumference 142–4
definition 141
diameter 141
π (pi) 142–4, 151
quizzes 156–7
radius 141, 144, 145
collecting like terms 120–1
common denominator 74
compass 141
compass (navigation) and measuring angles 126–7
composite numbers 45
cone, volume of 153
coordinates 182–7, 189, 190
and axes 183–6
finding distance between 184–5, 186
mnemonic to remember order of 184
quizzes 187
3-dimensional 185–6
corresponding angles 130
cosine/cos 176–7, 178
cube 161, 163, 164
cube root 96, 97
cubic meters 152
cubing 81
cuboids 152, 161
cumulative frequency table 201, 202
cylinders, volume of 152–3
decimal point 58–9
decimals 6, 57–65
adding 59–60
and fractions 70–1
multiplying 60–2
quizzes 65
recurring 71
rounding 63–4
subtracting 60
translating between percentages and 103–4
translating into standard form 92–3
degrees 125
denominators 68
common 74
Descartes, René 183
diameter (circle) 141
diamond 157
divisibility tests and multiples 46–7, 49
division 6, 35–43
carrying 39–41
chunking 37–8
and fractions 37, 77–8
long 41–2
and negative numbers 55
quizzes 43
remainders 36–7, 39–41
short 38–9
using times tables
backward 36
dodecahedron 162, 163, 164
equation of a line 190–2
equations 7, 112, 117–23
collecting like terms 120–1
and inequalities 7–8, 121–2
quizzes 123
equilateral triangles 133, 134, 136, 159
Euclid 125, 141
even numbers 46
as sum of two prime numbers 48
expanding brackets 114–15
exponential decay 82–3
exponential growth 81–2
factors 46
formulae, algebraic 111–12
fractional powers 97
fractions 58, 66–79
adding 74–6
canceling 76–7
converting into percentages 104
converting ratios into 106
and decimals 70–1
dividing 37, 77–8
multiplying 76–7
quizzes 72, 79
simplifying 68, 77
subtracting 76
top-heavy (improper) 69–70
frequency 201, 202
fundamental theorem of arithmetic 46
Goldbach’s conjecture 48
gradients 192–5
grams 89
graphs 188–96
equation of a line 190–2
negative gradients 194
plotting 191–2
quizzes 196
steepness and gradients 192–5
horizontal axis (x-axis) 183, 184, 185, 190
hypotenuse 167, 169, 176, 177
icosahedrons 163, 164
improper fractions 69–70
inequalities 7–8, 121–2
strict 121
weak 121
interest rates and percentages 106–7
interquartile range 202
inverse sine 180
irrational numbers 71, 142–3
isosceles trapezium 158
isosceles triangles 133, 134, 136
Jones, William 142
kilograms 89
kite 158
liters 89
and meters 152
logarithms 98–100
and calculators 98
quizzes 101
usefulness of 99–100
long division 41–2
long multiplication 27–9
Loomis, Elisha 170
mean 198–9
median 199, 201
megagram (ton) 90
and liters 152
metric system 88–90
milliliters 91
mixed numbers 69
mode 199–200, 201
multiples 46
and divisibility tests 46–7, 49
multiplication 6, 24–34
carrying 28–9
column method 32–3
and decimals 60–2
and fractions 76–7
grid method 31–2
long 27–9
and negative numbers 54–5
of powers 83–4
quizzes 34
by 10 amounts 30, 92
and times tables 26–7
negative gradient 194
negative numbers 19, 50–6
dividing 55
multiplying 54–5
and number line 51–3
quizzes 56
subtracting 53–4
uses of 52–3
negative powers of 10: 90
Newton 89
number line 51–3, 183
numerators 68
obtuse angles 126
obtuse triangles 134
octahedron 163, 164
odd numbers 46
opposite angles 129
outliers 198
parallel lines 129–30, 157, 195
parallelograms 158
areas 151
percentages 103–9
converting fractions into 104
increase and decrease 104–5
and interest rates 106–7
quizzes 108–9
translating between decimals and 103–4
percentiles 203
π (pi) 142–4, 151
pie charts 213–15, 217
plane 183
Platonic solids 162–4
polygons 159–61, 162
angles in regular 160–1
irregular 160
quizzes 165
regular 160, 162
powers 80–5, 99, 100
and calculators 81
exponential decay 82–3
exponential growth 81–2
fractional 97
and logarithms 98
multiplying 83–4
quizzes 85
and roots 96–7
and Sessa’s chessboard 81–2
of 10: 87–94
prefixes, metric 89–91
prime numbers 45–6, 48
breaking numbers down into 45–6, 48
and Goldbach’s conjecture 48
quizzes 49
probability 205–11
combining events “and” 207–8
combining events “or” 208–9
and counting successes 207
quizzes 210–11
proportion 104
protractor 127–8
pyramids 161
Pythagoras’ theorem 166–73, 179, 184–5, 186
Pythagorean triples 171
quadrilateral shapes 157–9
quartiles 202
radius (circle) 141, 144, 145
ratios 105–6
converting into fractions 106
rectangles 158
area 138, 150
recurring decimals 71
reflex angles 126, 159
reflex quadrilaterals 159
regular shapes 133
revolution 125–6
rhombus 157
right angle 126
right trapezium 158
right-angled triangles 134, 137
and Pythagoras’ theorem 166–71
roots 96–7, 98, 99
rounding 63–4
scalene triangles 133, 134
segmented bar charts 216–17
self-intersecting quadrilaterals 159
semicircle, area 151
Sessa’s chessboard 81–2
short division 38–9
π 142
> 7–8, 121
< 7–8, 121
≤ 8, 121
≥ 8, 121
simplifying algebra 113–14
simplifying fractions 68, 77
sine/sin 178, 180
inverse 180
SOH-CAH-TOA 178, 179
solids 161–4
area 153
volume 152
square numbers 27
square roots 96
squared units 149, 150, 152
squares 157, 159
area 144, 149, 150
squaring 81, 96
standard form 91–3
statistics 197–204
cumulative frequency table 201, 202
interquartile range 202
mean 198–9
median 199, 201
mode 199–200, 201
percentiles 203
quartiles 202
quizzes 204
strict inequalities 121
substitution and algebra 112–13
subtraction 18–23
borrowing 20–1
columns 20
and decimals 60
and fractions 76
and negative numbers 53–4
quizzes 23
rounding up and adding on 22
splitting numbers in your head 21–2
symbols 5–6
taking away see subtraction
tangent/tan 178, 179
multiplying by 30, 92
negative powers of 90–1
powers of 87–94
tetrahedron 162, 163, 164
times tables 26–7, 36
ton (megagram) 90
top-heavy fractions 69–70
trapeziums 158
isosceles 158
right 158
triangles 132–9
acute 134
angles in 133, 134–6
area of 136–8, 151
equilateral 133, 134, 136, 159
isosceles 133, 134, 136
obtuse 134
quizzes 139
right-angled 134, 137
scalene 133, 134
types of 133–4
within a rectangle 138
see also Pythagoras’ theorem; trigonometry
trigonometry 174–81
calculating angles of triangle from length 179–80
calculating length of triangle from angles 176–9
quizzes 181
units 88–9
unknowns 118
vertical axis (y-axis) 183, 184, 185, 190
volume 151–4
irregular shapes 154
quizzes 155
solid objects 152–3
weak inequalities 121
writing mathematics 6–7
x-axis 183, 185, 190
x-coordinate 190–1
y-axis 183, 185, 190
y-coordinate 190–1
y-intercept 195
z-axis 185–6